Methods and applications utilizing signal source memory space compression and signal processor computational time compression

ABSTRACT

A method and apparatus for a simplified approach for determining the output of a total covariance signal processor. A single set of offline calculations is performed and then used to estimate the output of the total covariance signal processor. A simplified approach for performing matrix inversion may also be used in determining the output of the total covariance processor.

This patent application claims the benefit of priority of U.S.Provisional Patent Application Ser. No. 60/799,696, filed May 10, 2006,entitled METHODS AND APPLICATIONS UTILIZING SIGNAL SOURCE MEMORY SPACECOMPRESSION AND SIGNAL PROCESSOR COMPUTATIONAL TIME COMPRESSION, theentire disclosure of which is incorporated herein by reference.

STATEMENT AS TO RIGHTS TO INVENTIONS MADE UNDER FEDERALLY SPONSOREDRESEARCH OR DEVELOPMENT

This application was supported in part by the Defense Advanced ResearchProjects Agency (DARPA) under the KASSPER Program Grant No.FA8750-04-1-004DARPA. The government of the United States may havecertain rights in this application

FIELD OF INVENTION

The present invention relates to signal processing, and in particular toefficient signal processing techniques which apply time compressionsolutions that increase signal processor throughput.

BACKGROUND OF THE INVENTION

In the systems arena two design problems prominently reign. One has asits fundamental goal the efficient storage of signals that are producedby a signal source of either artificial or biological origin, e.g.,voice, music, video and computer data sources. The other relates to theefficient processing of these signals that may for instance result intheir Fourier transform, covariance, etc. The design of efficient signalstorage algorithms relies heavily on source coding. The area of sourcecoding has a conspicuous recent history and has been one of the enablingtechnologies for what is known today as the information revolution. Thereason why this is the case is because source coding provides a soundpractical and theoretical measure for the information associated withany signal source output event and its average value or entropy. Thisinformation can then be used to provide an efficient replacement orsource coder for the signal source that can be either lossless or lossydepending if its output matches that of the signal source. Examples oflossless source coders are Huffman, Entropy, and Arithmetic coders asdescribed in The Communications Handbook, J. D. Gibson, ed., IEEE Press,1997. For the lossy case the standards of JPEG, MPEG and wavelets basedJPEG2000, predictive-transform (PT) source-coding, etc., have beenadvanced. See Predictive-Transform Source Coding with Bit Planes, Feriaand Licul, Submitted to 2006 IEEE Conference on Systems, Man andCybernetics, October 2006.

The design of efficient signal processing techniques is approached witha myriad of techniques that, unfortunately, are not similarly guided bya theoretical framework that encompasses both lossless and lossysolutions.

A real-world problem whose high performance is attributed to its use ofan intelligent system (IS) is knowledge-aided (KA) airborne movingtarget indicator (AMTI) radar such as found in DARPA's knowledge aidedsensory signal processing expert reasoning (KASSPER). The IS includestwo subsystems in cascade. The first subsystem is a memory devicecontaining the intelligence or prior knowledge. The intelligence isclutter whose knowledge facilitates the detection of a moving target.The clutter is available in the form of synthetic aperture radar (SAR)imagery where each SAR image requires 4 MB of memory space. Since therequired memory space for SAR imagery is prohibitive, it then becomesnecessary to use ‘lossy’ memory space compression source coding schemesto address this problem of memory space.

The second subsystem of the IS architecture is the intelligenceprocessor (IP) which is a clutter covariance processor (CCP). The CCP ischaracterized by the on-line computation of a large number of complexmatrices where a typical dimension for these matrices is 256×256 whichresults when both the number of antenna elements and transmitted antennapulses during a coherent pulse interval (CPI) is 16. Clearly thesecomputations significantly slow down the on-line derivation of thepre-requisite clutter covariances.

The present invention addresses these CCP computational issues using anovel time compression processor coding methodology that inherentlyarises as the ‘time compression dual’ of space compression sourcecoding. Further, missing from the art is a lossy signal processor thatutilizes efficient signal processing techniques to achieve high speedresults having a high confidence level of accuracy. The presentinvention can satisfy one or more of these and other needs.

In another aspect, the present invention relates to a simplifiedapproach for determining the output of a total covariance signalprocessor. Such an approach may be used, for example, in connection withan antenna-based radar system to make a decision as to whether or not atarget may be present at a particular location. Instead of estimatingthe output of a clutter covariance processor by performing certaincalculations offline, characterizing the input signal using onlinecalculations, and then using the online calculations to select one ofthe offline calculations, as discussed in the embodiments above dealingwith clutter covariance processors, in this embodiment, a single offlineset of calculations is performed and then used to estimate of the outputof the total covariance processor in conjunction with the antenna signalobtained at the time of viewing a target.

In yet another embodiment of the present invention, a simplifiedalgorithm for performing matrix inversion is used, for example, inconjunction with the previously described embodiment where the output ofthe total covariance processor is estimated using an inverse matrix,such as the inverse matrix R⁻¹ discussed above. The simplified matrixinversion algorithm utilizes a sidelobe canceller approach for matrixinversion, in conjunction with the predictive transform estimationapproaches discussed herein. The sidelobe canceller essentially removesand/or minimizes the effect of the antenna sidelobe signals on theantenna main beam return signal.

BRIEF DESCRIPTION OF THE DRAWING FIGURES

FIG. 1 depicts a conventional source coding system;

FIG. 2 depicts a processor coding system in accordance with anembodiment of the invention;

FIG. 3 depicts an embodiment of an intelligent system in accordance withthe present invention;

FIG. 4 depicts the duality of elements between the two complementarypillars of Compression-Designs.

FIG. 5 illustrates a block diagram of a KA-AMTI radar system;

FIG. 6 illustrates an embodiment of a Space-Time Processor in accordancewith the present invention;

FIG. 7 is a SAR image of the Mojave Airport, California;

FIG. 8 is the image of FIG. 7 averaged to yield 64 range bins;

FIG. 9 is plot of front clutter to noise ratio;

FIG. 10 depicts the structure of a radar-blind clutter coder inaccordance with the present invention;

FIG. 11 depicts the structure of a radar-seeing clutter coder inaccordance with the present invention;

FIG. 12 is a clutter cell centroid plot of the range bins depicted inFIG. 8;

FIG. 13 is an embodiment of a clutter covariance processor component inaccordance with the present invention;

FIGS. 14 a-14 d illustrate the simulation results for range bin #1 ofFIG. 8;

FIG. 15 depicts the antenna pattern of FIG. 5 in more detail;

FIGS. 16 a-16 b depicts plots of the average and maximum SINR versus therange bins of FIG. 8;

FIG. 17 depicts a 512 byte radar-blind PT decompressed SAR image;

FIG. 18 illustrates the RBCC clutter average power for range bin 1plotted versus clutter cell number;

FIG. 19 illustrates the average SINR error for all 64 range bins;

FIG. 20 illustrates the average SINR error versus range bin number forthe radar seeing case;

FIG. 21 illustrates the average SINR error versus range bin number forthe radar blind case;

FIG. 22 is a plot of SMI-AASE as a function of the ratio of SMI samples;

FIG. 23 is a block diagram of a PT source coder architecture;

FIG. 24 is a block diagram of a lossy PT encoder;

FIG. 25 is an illustration of transform pre-processing;

FIG. 26 is an illustration of predictive pre-processing;

FIG. 27 is a block diagram of a lossy PT decoder;

FIG. 28 is a block diagram of a lossless PT encoder;

FIG. 29 is an illustration of PT block decomposition;

FIG. 30 is an illustration of amplitude location decomposition;

FIG. 31 is an illustration of boundary decomposition;

FIG. 32 is an illustration of amplitude decomposition;

FIG. 33 is an illustration of magnitude decomposition;

FIG. 34 is a block diagram of a lossless PT decoder;

FIG. 35 is an illustration of zero rows construction;

FIG. 36 is an illustration of boundary bit plane construction;

FIG. 37 is an illustration of row ones construction;

FIG. 38 is an illustration of bit plane construction;

FIG. 39 is an illustration of a 512 byte JPEG2000 decompressed SARimage;

FIG. 40 is an illustration of SMI-AASE versus Lmsi/NM; and

FIG. 41 is a block diagram of an embodiment of the space-time sidelobecanceller structure for an antenna-based radar application according tothe present invention.

DETAILED DESCRIPTION OF THE ILLUSTRATIVE EMBODIMENTS

In FIG. 1 a source coding system is shown where the output of the signalsource is a discrete random variable X whose possible realizationsbelong to a finite alphabet of L elements, i.e., X ε{a₁, . . . , a_(L)}.Furthermore, the amount of “information” associated with the appearanceof the element a_(i) on the output of the signal source is denoted asI(a_(i)) and is defined in terms of the probability of a_(i), p(a_(i)),as follows:

$\begin{matrix}{{I\left( a_{i} \right)} = {\log_{2}\frac{1}{p\left( a_{i} \right)}}} & (1.1)\end{matrix}$

in units of bits (binary digits). Clearly from this expression it isnoted that a high probability event conveys a small amount ofinformation while one that rarely occurs conveys a lot of information.The source entropy is then defined as the average amount of informationin bits/sample H(X) that is associated with the random variable X. Thus

$\begin{matrix}{{H(X)} = {\sum\limits_{i = 1}^{L}{{p\left( a_{i} \right)}\log_{2}\frac{1}{p\left( a_{i} \right)}}}} & (1.2)\end{matrix}$

The signal source rate (in bits/sample) is defined by R_(ss) and isusually significantly greater than the source entropy H(X) as indicatedin FIG. 1. In the same figure, a source coder is presented which is madeup of an encoder followed by a decoder section. The input of the sourcecoder is the output of the signal source, while its output is anestimate {circumflex over (X)} of its input X. The source coder rate isdefined as R_(sc) and is generally smaller than the signal source rateR_(ss). The source coder will be lossless ({circumflex over (X)}=X) whenR_(sc) is greater than or equal to the source entropy H(X) and lossywhen R_(sc) is smaller that the source entropy as shown in FIG. 1.

A novel practical and theoretical framework, namely, processor coding,which arises as the time dual of source coding is part of the presentinvention. Processor coding directly addresses the problem of designingefficient signal processors. The aforementioned duality is apparent whenit is noted that the key concern of source coding is memory “spacecompression” while that of the novel processor coding methodology iscomputational “time compression.” Thus, both source coding and processorcoding solutions are noted to be characterized by compression designs,and thus, the combination of both coding design approaches is given thename compression-designs (“Conde”).

In viewing processor coding as the time dual of source coding, it isfirst realized that the time duals of bits, information, entropy, and asource coder in source coding are bors, latency, ectropy and a processorcoder in processor coding, respectively. These terms may be described asfollows:

1) “Bor” is short for a specified binary operator time delay;

2) “Latency” is the minimum time delay from the input to a specifiedscalar output of the signal processor that can be derived fromredesigning the internal structure of the signal processor subjected toimplementation components and architectural constraints;

3) “Ectropy”, with Greek roots ‘ec’ meaning outside and ‘tropy’ to look,is the maximum latency associated with all the scalar outputs of thesignal processor; and

4) “Processor coder” is the efficient signal processor that is derivedusing the processor coding methodology. A processor coder like a sourcecoder can be either lossless or lossy depending whether its outputmatches the original signal processor output.

In FIG. 2 a processor coding system is depicted where the output of thesignal processor is an M dimensional vector y=[y₁, . . . , y_(M)] andits input is the N dimensional vector x=[x₁, . . . , x_(N)].Furthermore, the amount of “latency” associated with the appearance ofthe element y_(i) on the output of the signal processor is denoted asL(y_(i)) and as mentioned earlier is the minimum time delay in timeunits of bors from the input x to scalar output y_(i) of the signalprocessor. The latency can then be derived from redesigning the internalstructure of the signal processor subjected to implementation componentsand architectural constraints. Clearly this definition implies the moresevere the implementation components and architectural constraints arethe larger the latency. These constraints are the time dual ofprobability in source coding when determining the amount of information.The ectropy of the signal processor G(y) or processor ectropy is thenthe maximum latency among all the M latency terms associate with the Melements of the signal processor output y, i.e.,

$\begin{matrix}{{G(y)} = {\max\limits_{L{(y_{i})}}\left\lbrack {{L\left( y_{1} \right)},\ldots \mspace{14mu},{L\left( y_{M} \right)}} \right\rbrack}} & (1.3)\end{matrix}$

The signal processor rate (in bors/y) is R_(SP) and is normallysignificantly greater than the processor ectropy G(y) as indicated inFIG. 2. In the same figure a processor coder is presented that is madeup of an encoder followed by a decoder section. The input of theprocessor coder x is the same as the input of the signal processor whileits output is an estimate ŷ of the signal processor output y. Theprocessor coder rate is R_(PC) and is smaller than the signal processorrate R_(SP). The processor coder will be lossless (ŷ=y) when R_(PC) isgreater than or equal to the processor ectropy and lossy when its R_(PC)is smaller than the processor ectropy as shown in FIG. 2.

The compression-designs or Conde methodology according to the presentinvention have been applied to a simulation of a real-world intelligentsystem problem with remarkable success. More specifically, themethodology has been applied to the design of a simulated efficientintelligent system for knowledge aided (KA) airborne moving targetindicator (AMTI) radar that is subjected to severely taxingenvironmental disturbances. The studied intelligent system includesclutter in the form of SAR imagery used as the intelligence or priorknowledge and a clutter covariance processor (CCP) used as theintelligence processor.

In FIG. 3 the basic structure of the intelligent system is shown andincludes a storage device for the clutter and the intelligence processorcontaining a clutter covariance processor receiving external inputs fromthe storage device as well as internal inputs. The internal inputs ofthe CCP are the antenna pattern and range bin geometry (APRBG) of theradar system and the complex clutter steering vectors. This intelligentsystem is responsible for the high signal to interference plus noiseratio (SINR) radar performance achieved with KA-AMTI but requiresprohibitively expensive storage and computational requirements. Theseproblems are addressed using the methodology of the present invention,Conde, with the following results:

1. For a “lossless” CCP coder to achieve outstanding SINR radarperformance, the source coder that replaces the clutter source should bedesigned with knowledge of the radar system APRBG: In other words thesource coder is radar seeing. This result yields a compression ratio of8,192 for the tested 4 MB SAR imagery but has the drawback of requiringknowledge about the radar system before the compression of the SAR imageis made.

2. For a significantly faster “lossy” CCP coder to derive exceptionalSINR radar performance the source coder that replaces the clutter sourcecan be designed without knowledge of the APRBG and is therefore said tobe radar blind. This result yields the same compression ratio of 8,192as the radar seeing case but is preferred since it is significantlysimpler to implement and can be used with any type of radar system.

The above two results indicate that the combination of universal, i.e.,radar blind, lossy source coders with an exceedingly fast lossy CCPcoder is the key to the derivation of truly efficient intelligentsystems for use in real-world radar systems and gives rise to thefollowing observations:

1. It suggests a paradigm shift in the design of efficient signalprocessors where the emphasis before was placed on the derivation oflossless efficient signal processors, such as a lossless Fast FourierTransform Processor, a lossless Fast Covariance Processor, etc., withoutany regard as to how the processor coder may be used in some particularapplication such as the target detection problem associated with radarsystems.

2. The outstanding SINR detection performance derived with highlycompressed prior knowledge, SAR imagery in the present invention,correlates quite well with how biological systems use highly compressedprior knowledge to make excellent decisions. Consider, for instance, howour brains expertly recognize a human face that had been viewed onlyonce before and could not be redrawn with any accuracy, based only onthis prior knowledge.

3. The duality that exists between space and time compressionmethodologies is pedagogically, theoretically, and practically appealingand their combined inner workings is extraordinary and worthy of notice.

4. It is of interest to note how the system performance remains high asboth the space and time compressions are increased, suggesting aninvariant-like property. As a fascinating and interesting practicalexample it should be noted that in physics there exists an observationframe of reference invariance that clearly constrains the evolution ofspace and time as it relates to the fact that the speed of light (inspace over time units) is measured to be the same in any observationframe.

FIG. 4 is a diagrammatic summary of the previously presentedobservations regarding certain characteristics of compression designs.First it is noted that FIG. 4 includes two columns. In the left column,space compression source coding is highlighted, while on the rightcolumn, time compression processor coding is illustrated. Nine differentcases are displayed in this image. CASE 0, appearing in the middle ofthe figure, displays the signal source and signal processor for whichone wishes to compress space and time, respectively. The picture in themiddle between the signal source and the signal processor is composed ofthree major parts, which are described as follows:

1) The sun triangles, consisting of eight different triangles, eachrepresent a different application where the signal source and signalprocessor may be used. The intensity of the shading inside thesetriangles denotes the application performance achieved in each case.Note that on the lower right hand side of the figure a chart is givensetting forth the triangle appearance and corresponding applicationperformance level. The darkest shading is used when an applicationachieves an optimum performance, whether or not the considered signalsource and signal processor are compressed. Clearly the application'sperformance is optimum and therefore the shading is darkest for thelossless signal source and signal processor of CASE 0;

2) The large gray colored circle without a highlighted boundaryrepresents the amount of memory space required to store the signaloutput of the signal source. On the left and bottom part of the image itis shown how the diameter of the gray colored circle decreases as therequired memory space decreases. Two cases are displayed. One casecorresponds to the lossless case and the other case corresponds to thelossy case. The lossy case in turn can be processor blind or processorseeing which displays an opening in the middle of the gray circle. Also,it should be noted that for the processor blind case the boundary of thegray circle is not smooth;

3) An unfilled black circle represents processor speed, where thereciprocal of its diameter reflects the time taken by the signalprocessor to produce an output. In other words the larger the diameterthe faster the processor. On the right and bottom part of the image twocases of time compression are displayed. First, the lossless case thathas smooth circles and then the lossy case that does not. CASE 1displays a “lossless” source coder using the signal processor of CASE 0where it is noted that the only difference between the illustrations forCASE 0 and CASE 1 is in the diameter of the space compression graycircle that is now smaller. CASE 2 is the opposite of CASE 1 where thediameter of the time compression unfilled black circle is now largersince the “lossless” processor coder is faster. CASE 3 combines CASES 1and 2 resulting in an optimum solution in all respects, except it maystill be taxing in terms of memory space and computational timerequirements. CASES 4 thru 8 are “lossy” cases. CASES 4 and 5 pertain toeither processor blind or processor seeing source coder cases where itis noted that the fundamental difference between the two is that theprocessor blind case yields a very poor application performance. On theother hand, the performance of the processor seeing case is suboptimumbut very close to the optimum one. CASE 6 addresses the “lossy”processor coder case in the presence of a “lossless” source coder. Forthis case, everything seems to be satisfactory except that the requiredmemory of the lossless source coder may still be too large. CASES 7 and8 present what occurs when the two types of lossy source coders are usedtogether with a ‘lossy’ processor coder. For these two cases it is foundthat the application performance is outstanding. CASE 7, in particular,is truly remarkable since it was found earlier for CASE 4 that aradar-blind source coder yields a very poor application performance whenthe processor coder is ‘lossless’. Thus it is concluded that CASE 7 ispreferred over all other cases since while achieving an outstandingapplication performance it is characterized by excellent space and timecompressions.

The time compression processor coding methodology gives rise to anexceedingly fast clutter covariance processor compressor (CCPC). TheCCPC includes a look up memory containing a very small number ofpredicted clutter covariances (PCCs) that are suitably designed off-line(e.g., in advance) using a discrete number of clutter to noise ratios(CNRs) and shifted antenna patterns (SAPs), where the SAPs aremathematical computational artifices not physically implemented. Theon-line selection of the best PCC is achieved by investigating for eachcase, e.g., each range bin, the actual CNR, as well as the clutter cellcentroid (CCC), which conveys information about the best SAP to select.The CCPC embodying the present invention is both very fast and yieldsoutstanding SINR radar performance using SAR imagery which is eitherradar-blind or radar-seeing and has been compressed by a factor of8,192. The radar-blind SAR imagery compression results are trulyremarkable in view of the fact that these simple and universal spacecompressor source coders cannot be used with a conventional CCP. Theadvanced CCPC is a ‘lossy’ processor coder that inherently arises from anovel practical and theoretical foundation for signal processing,namely, processor coding, that is the time compression signal processingdual of space compression source coding.

As described above, for processing coding the coding concepts includebor (or time delay needed for the execution of some specified binaryoperator), latency (or minimum time delay required to generate a scalaroutput for a signal processor after the internal structure of the signalprocessor has been redesigned subject to implementation components andarchitectural constraints), and ectropy (or maximum latency among allthe latencies derived for the signal processor scalar outputs),respectively.

FIG. 5 depicts an overview of a KA-AMTI radar system. It includes twomajor structures. They are: 1) An iso-range ring, or range bin, for auniform linear array (ULA) in uniform constant-velocity motion relativeto the ground. Only the front of the iso-range ring is shown,corresponding to angle displacements from −90° to 90° relative to theantenna array boresight; and 2) An AMTI radar composed of an antenna, aspace-time processor (STP) and a detection device. In KA-AMTI clutterreturns are available in the form of SAR imagery that is obtained from aprior viewing of the area of interest. From this figure it is alsonoticed that the range bin is decomposed into NC clutter cells. NC isoften greater than or equal to NM, where N is the number of antennaelements and M is the number of transmitted antenna pulses during acoherent pulse interval (CPI). In the example presented herein, M=16 andN=16. Table 1 is a summary of the relevant parameters, including thosefor M and N.

TABLE 1 a. Antenna N = 16, M = 16, d/λ = ½, f_(r) = 10³ Hz, f_(c) = 10⁹Hz, K^(f) = 4 × 10⁵ or 56 dBs, K^(b) = 10⁻⁴ or −40 dBs, b. Clutter N_(c)= 256, β = 1, 41 dBs < 10log₁₀CNR^(f) < 75 dBs, _(b) σ_(c,i) ² = 1 forall i, 10log₁₀CNR^(b) = −40 dBs, c. Target θ_(t) = 0° d. AntennaDisturbance σ_(n) ² = 1, θ_(AAM) = 2° e. Jammers N_(J) = 3, θ_(J) ₁ =−60°, θ_(J) ₂ = −30°, θ_(J) ₃ = 45°, 10log₁₀σ_(Ji) ² = 34 dBs for i = 1,2, 3, 10log₁₀JNR₁ = 53 dBs, 10log₁₀JNR₂ = −224 dBs and 10log₁₀JNR₃ = 66dBs f. Range Walk ρ = 0.999999 g. Internal Clutter Motion b = 5.7, ω =15 mph h. Narrowband CM ε_(i) = 0 for all i, γ_(i) for all i fluctuateswith a 5° rms i. Finite Bandwidth CM Δε = 0.001, Δφ = 0.1° j. AngleDependent CM B = 10⁸ Hz, Δθ = 28.6° k. Sample Matrix Inverse Lsmi = 8 ×64 = 512, σ_(diag) ² = 10

FIG. 6 illustrates an embodiment of the STP architecture in accordancewith the present invention. This input to the system is the addition oftwo signals, x and s. They are:

1) The NM×1 dimensional target steering vector s defined by

$\begin{matrix}{s = \left\lbrack {{{\underset{\_}{s}}_{1}\left( \theta_{t} \right)}\mspace{14mu} {{\underset{\_}{s}}_{2}\left( \theta_{t} \right)}\mspace{14mu} \ldots \mspace{14mu} {{\underset{\_}{s}}_{M}\left( \theta_{t} \right)}} \right\rbrack^{H}} & (2.1) \\{{{{\underset{\_}{s}}_{k}\left( \theta_{t} \right)} = {{^{j\; 2\; {\pi {({k - 1})}}{\overset{\_}{f}}_{D}^{t}}{{\underset{\_}{s}}_{I}\left( \theta_{t} \right)}\mspace{14mu} {for}\mspace{14mu} k} = 1}},\ldots \mspace{14mu},M} & (2.2) \\{{{\underset{\_}{s}}_{I}\left( \theta_{t} \right)} = \left\lbrack {{s_{1,1}\left( \theta_{t} \right)}\mspace{14mu} {s_{2,1}\left( \theta_{t} \right)}\mspace{14mu} \ldots \mspace{14mu} {s_{N,1}\left( \theta_{t} \right)}} \right\rbrack} & (2.3) \\{{{s_{k,1}\left( \theta_{t} \right)} = {{^{j\; 2\; {\pi {({k - 1})}}{\overset{\_}{\theta}}_{t}}\mspace{14mu} {for}\mspace{14mu} k} = 1}},\ldots \mspace{14mu},N} & (2.4) \\{{\overset{\_}{f}}_{D}^{t} = {f_{D}^{t}/f_{r}}} & (2.5) \\{f_{D}^{t} = {2\; {v_{t}/\lambda}}} & (2.6) \\{f_{r} = {1/T_{r}}} & (2.7) \\{{\overset{\_}{\theta}}_{t} = {\frac{d}{\lambda}{\sin \left( \theta_{t} \right)}}} & (2.8)\end{matrix}$

where: a) θ_(t) is the angle of attack (AoA) of the target with respectto boresight; b) d is the antenna inter-element spacing; c) λ is theoperating wavelength; d) θ _(t) is the normalized θ_(t); e) T_(r) is thepulse repetition interval (PRI); f) f_(r) is the pulse repetitionfrequency (PRF); g) v_(t) is the target radial velocity; h) f_(D) ^(t)is the Doppler of the target; and i) f ^(t) _(D) is the normalized f_(D)^(t).

2) The NM×1 dimensional vector x representing all system disturbances,which include the incident clutter, jammer, channel mismatch (CM),internal clutter motion (ICM), range walk (RW), antenna arraymisalignment (AAM), and thermal white noise (WN).

The NM×1 dimensional weight vector w, also shown in FIG. 6, multipliesthe STP input (s+x) yielding the STP generally complex scalar outputy=w^(H)(s+x). The expression for w is in turn given by the directinverse relation

w=R⁻¹s  (2.9)

that results from the maximization of the signal to interference plusnoise ratio (SINR)

SINR=w ^(H) ss ^(H) w/w ^(H) Rw  (2.10)

where the NM×NM dimensional matrix, R, is the total disturbancecovariance defined by R=E[xx^(H)]. To model this covariance thecovariance matrix tapers (CMTs) formulation was used resulting in

R={R _(C) O (R _(RW) +R _(ICM) +R _(CM))}+{R _(J) O R _(CM) }+R_(n)  (2.11)

R _(C) =R _(c) ^(f) +R _(c) ^(b)  (2.12)

where R_(n), R_(c) ^(f), R_(c) ^(b), R_(C), R_(J), R_(RW), R_(ICM) andR_(CM) are covariance matrices of dimension NM×NM and the symbol Odenotes a Hadamard product or element by element multiplication.Moreover, these disturbance covariances correspond to: R_(n) (thermalwhite noise); R_(c) ^(f) (front clutter); R_(c) ^(b) (back clutter);R_(C) (total clutter); R_(J) (jammer); R_(RW) (range walk); R_(ICM)(internal clutter motion); and R_(CM) (channel mismatch). Thecovariances R_(RW), R_(ICM) and R_(CM) are referred to as CMTs. Thecovariances R_(n) and R_(c) ^(f) are repeatedly used herein, and theyare described as follows:

Thermal white noise: R_(n) is described as follows

R_(n)=σ_(n) ²I_(NM)  (2.13)

where σ_(n) ² is the average power of thermal white noise and I_(NM) isan identity matrix of dimension NM×NM. Notice from Table 1, in theexamples presented herein, this noise power is assumed to be 1.

Front Clutter Covariance: R_(c) ^(f) is the output of the intelligentsystem of FIG. 6 and is described as follows:

$\begin{matrix}{R_{c}^{f} = {\sum\limits_{i = 1}^{N_{c}}{{p_{c}^{f}\left( {\theta_{c}^{i},\theta_{t}} \right)}{c^{f}\left( {\theta_{c}^{i},\theta_{AAM}} \right)}{c^{f}\left( {\theta_{c}^{i},\theta_{AAM}} \right)}^{H}}}} & (2.14) \\{{p_{c}^{f}\left( {\theta_{c}^{i},\theta_{t}} \right)} = {{G_{A}^{f}\left( {\theta_{c}^{i},\theta_{t}} \right)}_{f}\sigma_{c,i}^{2}}} & (2.15) \\{{G_{A}^{f}\left( {\theta_{c}^{i},\theta_{t}} \right)} = {K^{f}{\frac{\sin \left\{ {N\; \pi \frac{d}{\lambda}\left( {{\sin \left( \theta_{c}^{i} \right)} - {\sin \left( \theta_{t} \right)}} \right)} \right\}}{\sin \left\{ {\pi \frac{d}{\lambda}\left( {{\sin \left( \theta_{c}^{i} \right)} - {\sin \left( \theta_{t} \right)}} \right)} \right\}}}^{2}}} & (2.16) \\{{c^{f}\left( {\theta_{c}^{i},\theta_{AAM}} \right)} = \left\lbrack \begin{matrix}{{{}_{}^{}{c\_}_{}^{}}\left( {\theta_{c}^{i},\theta_{AAM}} \right)} \\{{{{}_{}^{}{c\_}_{}^{}}\left( {\theta_{c}^{i},\theta_{AAM}} \right)}\mspace{14mu} \ldots \mspace{14mu} {{{}_{}^{}{c\_}_{}^{}}\left( {\theta_{c}^{i},\theta_{AAM}} \right)}}\end{matrix}\mspace{14mu} \right\rbrack^{H}} & (2.17) \\{{{{{}_{}^{}{c\_}_{}^{}}\left( {\theta_{c}^{i},\theta_{AAM}} \right)} = {^{j\; 2\; {\pi {({k - 1})}}{{\overset{\_}{f}}_{D}^{c_{f}}{({\theta_{c}^{i},\theta_{AAM}})}}}{{\underset{\_}{c}}_{1}\left( \theta_{c}^{i} \right)}}}{{{{for}\mspace{14mu} k} = 1},\ldots \mspace{11mu},M}} & (2.18) \\{{{\underset{\_}{c}}_{1}\left( \theta_{c}^{i} \right)} = \left\lbrack {{c_{1,1}\left( \theta_{c}^{i} \right)}\mspace{14mu} {c_{2,1}\left( \theta_{c}^{i} \right)}\mspace{14mu} \ldots \mspace{14mu} {c_{N,1}\left( \theta_{c}^{i} \right)}} \right\rbrack} & (2.19) \\{{{c_{k,1}\left( \theta_{c}^{i} \right)} = {{^{j\; 2\; {\pi {({k - 1})}}{\overset{\_}{\theta}}_{c}^{i}}\mspace{14mu} {for}\mspace{14mu} k} = 1}},\ldots \mspace{11mu},N} & (2.20) \\{{{\overset{\_}{f}}_{D}^{c_{f}}\left( {\theta_{c}^{i},\theta_{AAM}} \right)} = {\beta \; {\overset{\_}{\theta}}_{c}^{i}\begin{Bmatrix}{{\cos \left( \theta_{AAM} \right)} +} \\\begin{matrix}{\sqrt{\begin{matrix}{{\sin^{2}\left( \theta_{AAM} \right)} +} \\{\left( {{\cos^{2}\left( \theta_{AMM} \right)} - 1} \right){\sin^{2}\left( \theta_{c}^{i} \right)}}\end{matrix}}/} \\{\sin \left( \theta_{c}^{i} \right)}\end{matrix}\end{Bmatrix}}} & (2.21) \\{\beta = \frac{v_{p}T_{r}}{d/2}} & (2.22) \\{\theta_{c}^{- i} = {\frac{d}{\lambda}{\sin \left( \theta_{c}^{i} \right)}}} & (2.23)\end{matrix}$

where: a) the index i refers to the i-th front clutter cell on the rangebin section shown on FIG. 5; b) σ_(c) ^(i) is the AoA of the i-thclutter cell; c) θ_(AAM) is the antenna array misalignment angle; d)_(f)σ_(c,i) ² is the i-th front clutter source cell power (excluding theantenna gain); e) G_(A) ^(f)(θ_(c) ^(i),θ_(t)) is the antenna patterngain associated with the i-th front clutter cell; f) K^(f) is the frontglobal antenna gain; g) p_(c) ^(f)(θ_(c) ^(i),θ_(t)) is the “total” i-thfront clutter cell power. In the example presented herein, the 4 MB1,024 by 254 samples SAR image of the Mojave Airport in California (FIG.7) will be used with groups of sixteen consecutive rows averaged toyield the 64 range bins, as depicted in FIG. 8. In FIG. 9 a plot is alsogiven of the front clutter to noise ratio (CNR^(f)), i.e.,

$\begin{matrix}{{{CNR}^{f} = {{{R_{c}^{f}\left( {1,1} \right)}/\sigma_{n}^{2}} = {\sum\limits_{i = 1}^{N_{c}}{{p_{c}^{f}\left( {\theta_{c}^{i},\theta_{t}} \right)}/\sigma_{n}^{2}}}}},} & (2.24)\end{matrix}$

for the 64 range bins with values ranging from 41 to 75 dBs where theaverage power of the thermal white noise was assumed equal to 1, i.e.,σ_(n) ²=1; h) c^(f)(θ_(c) ^(i),θ_(AAM)) is the NM×1 dimensional andcomplex i-th clutter cell steering vector; i) v_(p) is the radarplatform speed; j) θ_(c) ^(−i) is the normalized θ_(c) ^(i); and k) β isthe ratio of the distance traversed by the radar platform during thePRI, v_(p)T_(r), to the half antenna inter-element distance, d/2.

At this point it should be noted that expressions (2.14)-(2.15) definethe clutter covariance processor or intelligence processor of theintelligent system of FIG. 3. In addition, the front clutter source cellpower is the output of the intelligence source that the intelligenceprocessor operates on.

Back Clutter

The back clutter covariance R_(c) ^(b) is given by

$\begin{matrix}{R_{c}^{b} = {\sum\limits_{i = 1}^{N_{c}}{{p_{c}^{b}\left( {\theta_{c}^{i},\theta_{t}} \right)}{c^{b}\left( {\theta_{c}^{i},\theta_{AAM}} \right)}{c^{b}\left( {\theta_{c}^{i},\theta_{AAM}} \right)}^{H}}}} & (2.25) \\{{p_{c}^{b}\left( {\theta_{c}^{i},\theta_{t}} \right)} = {{G_{A}^{b}\left( {\theta_{c}^{i},\theta_{t}} \right)}_{b}\sigma_{c,i}^{2}}} & (2.26) \\{{G_{A}^{b}\left( {\theta_{c}^{i},\theta_{t}} \right)} = {K^{b}{\frac{\sin \left\{ {N\; \pi \frac{d}{\lambda}\left( {{\sin \left( \theta_{c}^{i} \right)} - {\sin \left( \theta_{t} \right)}} \right)} \right\}}{\sin \left\{ {\pi \frac{d}{\lambda}\left( {{\sin \left( \theta_{c}^{i} \right)} - {\sin \left( \theta_{t} \right)}} \right)} \right\}}}^{2}}} & (2.27) \\{{c^{b}\left( {\theta_{c}^{i},\theta_{AAM}} \right)} = \left\lbrack \begin{matrix}{{{}_{}^{}{c\_}_{}^{}}\left( {\theta_{c}^{i},\theta_{AAM}} \right)} \\{{{{}_{}^{}{c\_}_{}^{}}\left( {\theta_{c}^{i},\theta_{AAM}} \right)}\mspace{14mu} \ldots \mspace{14mu} {{{}_{}^{}{c\_}_{}^{}}\left( {\theta_{c}^{i},\theta_{AAM}} \right)}}\end{matrix}\mspace{14mu} \right\rbrack^{H}} & (2.28) \\{{{{{}_{}^{}{c\_}_{}^{}}\left( {\theta_{c}^{i},\theta_{AAM}} \right)} = {^{j\; 2\; {\pi {({k - 1})}}{{\overset{\_}{f}}_{D}^{c_{b}}{({\theta_{c}^{i},\theta_{AAM}})}}}{{\underset{\_}{c}}_{1}\left( \theta_{c}^{i} \right)}}}{{{{for}\mspace{14mu} k} = 1},\ldots \mspace{11mu},M}} & (2.29) \\{{{\overset{\_}{f}}_{D}^{c_{b}}\left( {\theta_{c}^{i},\theta_{AAM}} \right)} = {\beta \; {\overset{\_}{\theta}}_{c}^{i}\begin{Bmatrix}{{\cos \left( \theta_{AAM} \right)} -} \\\begin{matrix}{\sqrt{\begin{matrix}{{\sin^{2}\left( \theta_{AAM} \right)} +} \\{\left( {{\cos^{2}\left( \theta_{AMM} \right)} - 1} \right){\sin^{2}\left( \theta_{c}^{i} \right)}}\end{matrix}}/} \\{\sin \left( \theta_{c}^{i} \right)}\end{matrix}\end{Bmatrix}}} & (2.30)\end{matrix}$

where: a) the index i now refers to the i-th clutter cell on the backside of the iso-range ring, not shown in FIG. 5; b) θ_(c) ^(i) is theAoA of the i-th back clutter cell; c) _(b)σ_(c,i) ² is the i-th backclutter source cell power (assumed one for all i; see Table 1, entry b);d) G_(A) ^(b)(θ_(c) ^(i),θ_(t)) is the back antenna pattern gainassociated with _(b)θ_(c,i) ²; e) K^(b) is the global back antenna gain(assumed 10⁻⁴, see Table 1, entry a); f) p_(c) ^(b)(θ_(c) ^(i),θ_(t)) isthe total clutter cell power of the i-th back clutter cell (the backclutter to noise ratio, CNR^(b), is described as:

$\begin{matrix}{{{CNR}^{b} = {{{R_{c}^{b}\left( {1,1} \right)}/\sigma_{n}^{2}} = {\sum\limits_{i = 1}^{N_{c}}{{p_{c}^{f}\left( {\theta_{c}^{i},\theta_{t}} \right)}/\sigma_{n}^{2}}}}},} & (2.31)\end{matrix}$

and is assumed to be −40 dB, see Table 1, entry b); f) c^(b)(θ_(c)^(i),θ_(AAM)) is the NM×1 dimensional and complex steering vectorassociated with _(b)σ_(c,i) ²; and g) c ₁(θ_(c) ^(i)) is as defined in(2.19)-(2.20).

Jammer

The jammer covariance R_(J) is given by

$\begin{matrix}{R_{J} = {\sum\limits_{i = 1}^{N_{J}}{{p_{J}\left( {\theta_{J}^{i},\theta_{t}} \right)}\left( {I_{M} \otimes 1_{N \times N}} \right){O\left( {{j\left( \theta_{J}^{i} \right)} \cdot {j\left( \theta_{J}^{i} \right)}^{H}} \right)}}}} & (2.32) \\{{p_{J}\left( {\theta_{J}^{i},\theta_{t}} \right)} = {{G_{A}^{f}\left( {\theta_{J}^{i},\theta_{t}} \right)}\sigma_{J,i}^{2}}} & (2.33) \\{{j\left( \theta_{J}^{i} \right)} = \left\lbrack {{j_{1}\left( \theta_{J}^{i} \right)}\mspace{14mu} {j_{2}\left( \theta_{J}^{i} \right)}\mspace{14mu} \ldots \mspace{14mu} {j_{M}\left( \theta_{J}^{i} \right)}} \right\rbrack^{H}} & (2.34) \\{{{j_{k}\left( \theta_{J}^{i} \right)} = {{{j_{1}\left( \theta_{J}^{i} \right)}\mspace{14mu} {for}\mspace{14mu} k} = 1}},\ldots \mspace{14mu},M} & (2.35) \\{{j_{1}\left( \theta_{J}^{i} \right)} = \left\lbrack {{j_{1,1}\left( \theta_{J}^{i} \right)}\mspace{14mu} {j_{2,1}\left( \theta_{J}^{i} \right)}\mspace{14mu} \ldots \mspace{14mu} {j_{N,1}\left( \theta_{J}^{i} \right)}} \right\rbrack^{H}} & (2.36) \\{{{j_{k,1}\left( \theta_{J}^{i} \right)} = {{^{j\; 2\; \pi \; {({k - 1})}{\overset{\_}{\theta}}_{J}^{i}}{for}\mspace{14mu} k} = 1}},\ldots \mspace{14mu},N} & (2.37) \\{{\overset{\_}{\theta}}_{J}^{i} = {\frac{d}{\lambda}{\sin \left( \theta_{J}^{i} \right)}}} & (2.38)\end{matrix}$

where: a) the index i refers to the i-th jammer on the range bin; b)N_(J) is the total number of jammers (assumed to be three; see Table 1,entry e); c) θ_(J) ^(i) is the AoA of the i-th jammer (the location ofthe three assumed jammers are at −60°, −30°, and 45°; see Table 1, entrye); d)

is the Kronecker (or tensor) product; e) I_(M) is an identity matrix ofdimension M by M; f) 1_(N×N) is a unity matrix of dimension N by N; g)σ_(J) _(J) ² is the i-th jammer power (34 dB is assumed for the threejammers considered; see Table 1, entry e); h) p_(J)(θ_(c) ^(i),θ^(t)) isthe “total” i-th jammer power, the jammer to noise ratio (JNR), isdescribed as follows:

$\begin{matrix}{{{JNR} = {{{R_{J}\left( {1,1} \right)}/\sigma_{n}^{2}} = {\sum\limits_{i = 1}^{N_{J}}{{p_{J}\left( {\theta_{c}^{i},\theta_{t}} \right)}/\sigma_{n}^{2}}}}},} & (2.39)\end{matrix}$

is given by 53, −224, and 66 dB for the jammers at −60°, −30°, and 45°,respectively; see Table 1, entry e); and i) j(θ_(J) ^(i)) is the NM×1dimensional and complex i-th jammer steering vector that is noted fromthe defining equations (2.34)-(2.38) to be Doppler independent.

Range Walk

The range walk or RW CMT, R_(RW), is described as follows:

R_(RW)=R_(RW) ^(time)

R_(RW) ^(space)  (2.40)

[R _(RW) ^(time)]_(i,k)=ρ^(|i-k|)  (2.41)

R _(RW) ^(space)=1_(N×N)  (2.42)

ρ=ΔA/A=ΔA/{ΔRΔθ}=ΔA/{(c/B)Δθ}  (2.43)

where: a) c is the velocity of light; b) B is the bandwidth of thecompressed pulse; c) ΔR is the range-bin radial width; d) Δθ is themainbeam width; e) A is the area of coverage on the range bin associatedwith Δθ at the beginning of the range walk; f) ΔA is the remnants ofarea A after the range bin migrates during a CPI; and g) ρ is thefractional part of A that remains after the range walk (for example,ρ=0.999999; see Table 1, entry f).

Internal Clutter Motion

The internal clutter motion or ICM CMT, R_(ICM), is described asfollows:

$\begin{matrix}{R_{ICM} = {R_{ICM}^{time} \otimes R_{ICM}^{space}}} & (2.44) \\{\left\lbrack R_{ICM}^{time} \right\rbrack_{i,k} = {\frac{r}{r + 1} + {\frac{1}{r + 1}\frac{({bc})^{2}}{({bc})^{2} + \left( {4\; \pi \; f_{c}{{k - i}}T_{r}} \right)^{2}}}}} & (2.45) \\{R_{ICM}^{space} = 1_{N \times N}} & (2.46) \\{{10\; \log_{10}r} = {{{- 15.5}\; \log_{10}\omega} - {12.1\; \log_{10}f_{c}} + 63.2}} & (2.47)\end{matrix}$

where: a) f_(c) is the carrier frequency in megahertz; b) ω is the windspeed in miles per hour; c) r is the ratio between the dc and ac termsof the clutter Doppler power spectral density; d) b is a shape factorthat has been tabulated; and e) c is the speed of light. In the examplepresented herein, f_(c)=1,000 MHz, ω=15 mph and b=5.7; see Table 1,entries a, g.

Channel Mismatch

The total channel mismatch or CM CMT, R_(CM), is described as follows:

R_(CM)=R_(NB)O R_(FB)O R_(AD)  (2.48)

where R_(NB), R_(FB) and R_(AD) are composite CMTs, as described below.

Angle Independent Narrowband

R_(NB) is an angle-independent narrowband or NB channel mismatch CMT,which is described as follows:

$\begin{matrix}{R_{NB} = {qq}^{H}} & (2.49) \\{q = \left\lbrack {{\underset{\_}{q}}_{1}\mspace{14mu} {\underset{\_}{q}}_{2}\mspace{14mu} \ldots \mspace{14mu} {\underset{\_}{q}}_{M}} \right\rbrack^{H}} & (2.50) \\{{{\underset{\_}{q}}_{k} = {{{\underset{\_}{q}}_{1}\mspace{14mu} {for}\mspace{14mu} k} = 1}},\ldots \mspace{14mu},M} & (2.51) \\{{\underset{\_}{q}}_{1} = \left\lbrack {ɛ_{1}^{j\; \gamma_{1}}ɛ_{2}^{j\; \gamma_{2}}\mspace{14mu} \ldots \mspace{14mu} ɛ_{N}^{j\; \gamma_{N}}} \right\rbrack} & (2.52)\end{matrix}$

where in (2.52) Δε₁, . . . , Δε_(N) and Δγ₁, . . . , Δγ_(N) denoteamplitude and phase errors, respectively. In the example presentedherein, the amplitude errors are assumed to be zero and the phase errorsare assumed to fluctuate with a 5° root mean square (rms); see Table 1,entry h.

Finite Bandwidth

R_(FB) is a finite (nonzero) bandwidth or FB channel mismatch CMT, whichis described as follows:

$\begin{matrix}{R_{FB} = {R_{FB}^{time} \otimes R_{FB}^{space}}} & (2.53) \\{R_{FB}^{time} = 1_{M \times M}} & (2.54) \\{\left\lbrack R_{FB}^{space} \right\rbrack_{i,k} = {{\left( {1 - {{\Delta ɛ}/2}} \right)^{2}\sin \; {c^{2}\left( {{\Delta\varphi}/2} \right)}\mspace{14mu} {for}\mspace{14mu} i} \neq k}} & (2.55) \\{{\left\lbrack R_{FB}^{space} \right\rbrack_{i,i} = {{1 - {\Delta ɛ} + {\frac{1}{3}{\Delta ɛ}^{2}\mspace{20mu} {for}\mspace{14mu} i}} = 1}},\ldots \mspace{14mu},N} & (2.56)\end{matrix}$

where in (2.55)-(2.56) Δε and Δφ denote the peak deviations ofdecorrelating random amplitude and phase channel mismatch, respectively.In the example presented herein, Δε=0.001 and Δφ=0.1°; see Table 1,entry i.

Angle Dependent

R_(AD) is a reasonably approximate angle-independent CMT forangle-dependent or AD channel mismatch, which is described as follows:

$\begin{matrix}{R_{AD} = {R_{AD}^{time} \otimes R_{AD}^{space}}} & (2.57) \\{R_{AD}^{time} = 1_{M \times M}} & (2.58) \\{\left\lbrack R_{AD}^{space} \right\rbrack_{i,k} = {{\sin \; {c\left( {B{{k - i}}\frac{d}{c}\sin \; ({\Delta\theta})} \right)}\mspace{14mu} {for}\mspace{14mu} i} \neq k}} & (2.59) \\{\left\lbrack R_{AD}^{space} \right\rbrack_{i,i} = 1} & (2.60)\end{matrix}$

where B is the bandwidth of an ideal bandpass filter and Δθ is asuitable measure of mainbeam width. In the example presented herein,B=100 MHz and Δθ=28.6°; see Table 1, entry j.

Optimum Direct Inverse

The w that maximizes the SINR expression (2.10) is given by thefollowing expression:

w=R⁻¹s.  (2.61)

Two general approaches can be used to derive R. They are:

1) The first approach is not knowledge-aided and is given by the SMIexpression:

$\begin{matrix}{{\,^{smi}R} = {{\frac{1}{Lsmi}{\sum\limits_{i = 1}^{Lsmi}{X_{i}X_{i}^{H}}}} + {\sigma_{diag}^{2}I}}} & (2.62)\end{matrix}$

where X_(i) denotes radar measurements from range bins close to therange bin under investigation, Lsmi is the number of measurement samplesand σ² _(diag)I is a diagonal loading term. X_(i) may be derived via thefollowing generating expression

X _(i) =R _(i) ^(−1/2) x _(i)  (2.63)

where: a) x_(i) is a zero mean, unity variance, NM dimensional complexrandom draw; and b) R_(i) is the total disturbance covariance(2.11)-(2.12) associated with the i-th range bin. In the examplepresented herein σ_(diag) ²=10σ_(n) ²=10; see Table 1, entry k.

2) The second approach is KA and assumes knowledge of all thecovariances associated with the total disturbance covariance R, see(2.11)-(2.12).

Radar Blind and Radar Seeing Source-Coders

FIG. 3 presents the intelligence source and intelligence processorsubsystems of the intelligent system of FIG. 6. The intelligence sourcecontains the stored SAR imagery or clutter, while theintelligence-processor or CCP uses as its external input the output ofthe SAR imagery source, and as internal inputs the antenna pattern andrange bin geometry or APRBG and the front clutter steering vectors(2.18) to compute the front clutter covariance matrix (2.15). Althoughthis system results in optimum SINR radar performance, it is highlyinefficient in terms of both its memory storage and on-line computinghardware requirements. To alleviate the memory storage problemassociated with the intelligent system two different types of sourcecoders may be investigated as tentative replacements for theintelligence-source of FIG. 3. These include a simplepredictive-transform (PT) radar-blind scheme that is oblivious to theAPRBG and a more elaborate radar-seeing scheme that makes use of theAPRBG. These two schemes are now reviewed in the form of block diagramdescriptions.

FIG. 10 illustrates the basic structure of a radar-blind clutter coder(RBCC), which includes an intelligence source coder containing thecompressed or encoded clutter where the APRBG was not used. Oneadvantage of a radar-blind clutter coder is that the compressed cluttercan be used with any kind of AMTI radar system without regard to theactual APRBG environment. A clutter decompressor is included to derivean estimate for the uncompressed clutter for use by the conventional CCPor intelligence processor. The combination of the RBCC and conventionalCCP is denoted here as RBCC-CCP for short. It has been found that thissimple scheme generally does not produce a satisfactory SINR radarperformance with reasonable compression ratios for SAR imagery.

FIG. 11 depicts the radar-seeing clutter coder (RSCC) structure, wherethe only difference from that of the radar-blind case of FIG. 10 is thatthe source-coder makes use of the APRBG. The combination of the RSCC anda conventional CCP is denoted as RSCC-CCP for short. It has been foundthat outstanding SINR radar performance is derived when SAR imagery iscompressed from 4 MB to 512 bytes for a compression ratio of 8,192. TheRSCC scheme requires that minimum and maximum CNR values be found forthe SAR image when processed in any direction. In the example presentedherein, 41 and 75 dB were used for these values, respectively, which arealso noted to be in accord with the CNR plot of FIG. 9. Using theseextreme CNR values, the front clutter source cell power _(f)σ_(c,i) ²was generally in the range between 0.0077 and 7.7 which correspond tothe minimum and maximum CNR values of 41 and 75 dB, respectively, aswell as the assumed front global antenna gain given in Table 1. Theresultant power limited SAR image was then compressed using standardcompression schemes, e.g., PT source-coding.

In FIG. 17 a 512 byte radar-seeing PT decompressed SAR image is shownfor a compression ratio of 8,192. In FIG. 20 the corresponding averageSINR error is given for the 64 range-bins of FIG. 8. Note that thisfigure is characterized by a very small AASE value of approximately 0.7dB. A comparison of FIG. 20 and FIG. 21 reveals that the radar-seeingscheme achieves much better SINR radar performance for the same amountof compression. However, it should be kept in mind, that thisimprovement is achieved at the expense of the prerequisite priorknowledge of the APRBG.

The clutter covariance processor compressor (CCPC) embodying the presentinvention achieves significant “on-line” (i.e., real time) computationaltime compression over the conventional clutter covariance processor orCCP. Simulations have shown that the CCPC is in fact the timecompression dual of a space compression “lossy” source coder. The CCPCaccording to the present invention is eminently lossy since its outputdoes not need to emulate that of the straight CCP. This is the casesince its stated objective is to derive outstanding SINR radarperformance regardless of how well its output compares with that of thelocal intelligence processor. It should be noted that the computationalburden or time delay of the conventional CCP describing equations(2.14)-(2.15) is governed by the need to determine “on-line” the frontclutter steering matrix Nc times, where each of these NM×NM dimensionalmatrices is weighted by the scalar and real cell power p_(c) ^(f)(θ_(c)^(i),θ_(t)).

Furthermore, from expression (2.15) it is noted that the shape of therange bin cell power is a function of the antenna pattern G_(A)^(f)(θ_(c) ^(i),θ_(t)) as well as the front clutter source cell power_(f)σ_(c,i) ² which often varies drastically from range bin to rangebin. Clearly, the variation of the clutter source cell power _(f)σ_(c,i)² from range bin to range bin is the source of the on-line computationalburden associated with (2.14)-(2.15) since otherwise these expressionscould have been solved off-line.

The on-line computational time delay problem of the conventional CCP isaddressed in two steps, where each step has two parts.

Step I:

Part I.A External CCP Input: In this first part, a simple mathematicalmodel for the external input of the CCP is sought. This external inputis the clutter source cell power waveform {_(f)σ_(c,i) ²} and itsmathematical model is selected to be the power series

K₀+K₁i+K₂i²+ . . . ,  (3.1)

where K_(j) for all j are real constants that are determined on-line foreach range bin using as a basis the measured input waveform {_(f)σ_(c,i)²}. Since a desirable result is to achieve the smallest possible“on-line” computational time delay while yielding a satisfactory SINRradar performance, a single constant, K₀, has been selected to model theentire clutter source cell power waveform. The numerical value for K₀ isdetermined such that it reflects the strength of the clutter. Thestrength of the clutter, in turn, is related to the front clutter tonoise ratio or CNR^(f) defined earlier in (2.24) and plotted in FIG. 9for the 64 range bins of FIG. 8. The CNR^(f) will be one of two real andscalar values derived by the CCPC where it is assumed that the thermalwhite noise variance σ_(n) ² is 1.

Part I.B Internal CCP Input: In this second and last part of Step I, asuitable modulation of the antenna pattern waveform {G_(A) ^(f)(θ_(c)^(i),θ_(t)} is sought. The modulation of this internal CCP input can beachieved in several ways. Two of them are: a) By using peak-modulationwhich consists of shifting the peak of the antenna pattern to somedirection away from the target; and b) By using antennaelements-modulation which consists of widening or narrowing the antennapattern mainbeam by modifying the number of “assumed” antenna elementsN. It is emphasized here that these are only a mathematical alterationof the antenna pattern, since the true antenna pattern remainsunaffected. Peak-modulation may be selected since, as mentioned earlier,the main objective is to achieve the smallest possible “on-line”computational delay for the computational time compressed CCP.Furthermore, to find the position to where the peak of the antennapattern should be shifted to, the clutter cell centroid (CCC) or centerof mass of the clutter is evaluated for each range bin. The CCC is thesecond of two scalar values derived by the CCPC and is given by thefollowing expression

$\begin{matrix}{{CCC} = {\left. {\sum\limits_{i = 1}^{N_{C}}{{i\left( {{G_{A}^{f}\left( {\theta_{c}^{i},\theta_{i}} \right)}_{f}\sigma_{c,i}^{2}} \right)}/{CNR}^{f}}} \middle| \sigma_{n}^{2} \right. = 1}} & (3.2)\end{matrix}$

In FIG. 12 the CCC plot is shown for the 64 range bins of the SAR imagegiven in FIG. 8. It should be noted that for many of the 64 range binsthe CCC varies significantly from the position of the assumed target at128.5 (0° from boresight). Clearly, for the isotropic clutter case theCCC will reside at boresight.

Step II

Part II.A Off-Line Evaluations: In this first part of Step II a finiteand fixed number of predicted clutter covariances or PCCs are foundoff-line. This is accomplished using the CCP describing equations(2.14)-(2.15) subject to the simple clutter model (3.1) and a modulatedantenna pattern which results in a small and fixed number of highlylossy clutter covariance realizations. The PCCs are derived from thefollowing expressions:

$\begin{matrix}{{{{PCC}\left( {k,j} \right)} = {\sum\limits_{i = 1}^{N_{C}}{{p_{pc}^{f}\left( {\theta_{c}^{i},\theta_{t},\theta^{k},{PCNR}_{j}} \right)}{c^{f}\left( {\theta_{c}^{i},\theta_{A}} \right)}{c^{f}\left( {\theta_{c}^{i},\theta_{A}} \right)}^{H}}}}{{k = 1},\ldots \mspace{14mu},{{{N_{SAP}\&}\mspace{14mu} j} = 1},\ldots \mspace{14mu},N_{CNR}}} & (3.3) \\{{p_{pc}^{f}\left( {\theta_{c}^{i},\theta_{t},\theta^{k},{PCNR}_{j}} \right)} = {{G_{A}^{f}\left( {{\theta_{c}^{i} - \theta^{k}},\theta_{t}} \right)}{K_{0}\left( {PCNR}_{j} \right)}}} & (3.4) \\{{PCNR}_{j} \in \left\lbrack {{PCNR}_{Min},\ldots \mspace{14mu},{PCNR}_{Max}} \right\rbrack} & (3.5)\end{matrix}$

where: a) p_(pc) ^(f)(.) is the predicted front clutter power; b) G_(A)^(f)(θ_(c) ^(i)−θ^(k),θ_(t)) is a shifted antenna pattern or SAP wherethe peak value of the actual antenna pattern (2.16) has been shiftedfrom θ_(c) ^(i)=θ_(t) to θ_(c) ^(i)=θ_(t)+θ^(k); c) θ^(k) denotes theamount of angular shift of the SAP away from the assumed target positionθ_(t) (the SAPs are generally designed in pairs, one associated withθ^(k) and the other with −θ^(k)); d) N_(SAP) is the number of SAPsconsidered (in the simulations the cases with N_(SAP)=1, 3 and 5 will beconsidered); e) PCNR_(j) is the j-th predicted CNR value; e)K₀(PCNR_(j)) is the PCC constant gain that gives rise to the PCNR_(j);f) NCNR is the number of assumed PCNR values (predicted clutter to noiseratio) in the example presented herein, N_(CNR)=2); and f) PCNR_(Min)and PCNR_(Max) are minimum and maximum PCNR values, respectively,suitably evaluated for each SAR image (these values are 57 and 75 dB,respectively, for the SAR image presented herein).In FIG. 13 the previously described CCPC is shown for the case where sixpredicted clutter covariances or PCCs are used. These PCCs were derivedassuming three SAPs and two PCNRS. The SAPs were shifted to −7° (cell118 on the range bin), 0° (128.5) and 7° (139) from boresight and thePCNRs were 57 and 75 dB, respectively. The CCPC includes CNR and CCCprocessors where their input is given by the waveform {_(f) ^(X)σ_(c,i)²} and _(f) ^(X)σ_(c,i) ² denotes the i-th front clutter source cellpower corresponding to three different cases for X, which are asfollows:

1. X=UCMD when the clutter emanates from the storage uncompressedclutter memory device (UCMD) of FIG. 3.

2. X=RBCC when the clutter is generated from the radar-blind cluttercoder of FIG. 10.

3. X=RSCC when the clutter is derived from the radar-seeing cluttercoder of FIG. 11.

After the CNR and CCC values are determined, the CCPC selects from thememory containing the 6 PCCs of FIG. 13 the one that is better matchedto the measured CCC and CNR processor output values. For instance, ifthe CCC processor output is 140, the selection process is narrowed downto the pair of PCCs that were evaluated using the SAP that is shifted toposition 139 on the range bin (or +7°), since it is the closest. Inaddition, if the measured CNR processor output is 60 dB the element ofthe selected PCC pair associated with the 75 dB PCNR is selected. Itshould be noted that the PCNRj selected is the one “above” the measuredCNR processor output value.

At this point two observations are made. The first is that the Centroidand CNR Processors of FIG. 13 govern the time delay associated with theCCPC, and thus constitute a ‘lossy processor encoder’ since they encodein a lossy fashion the time delay essence, i.e., the ectropy, of theoriginal CCP. The second observation is that the look up memory sectionof FIG. 13 is a ‘lossy processor decoder’ since it reconstructs a highlylossy version of the output of the original CCP.

Three space-time processors or SPTs are now described where the contentof the UCMD is applied to three different types of CCPCs. The weightingvector w of the three STPs is described as follows

$\begin{matrix}{w = {\left\lbrack {\,_{CCPC}^{UCMD}R} \right\rbrack^{- 1}s}} & (3.6) \\{{\,_{CCPC}^{UCMD}R} = \left. R \right|_{R_{c}^{f} = {{}_{}^{}{}_{}^{}}}} & (3.7) \\\left. {{{}_{}^{}{}_{}^{}} \in \left\{ {{PCC}\left( {k,j} \right)} \right\}} \right|_{S_{CCPC}^{UCMD}} & (3.8)\end{matrix}$

where: a)

R|_(R_(c)^(f) = )

is the total disturbance covariance (2.11)-(2.12) with the CCPC outputof FIG. 13, _(CCPC) ^(UCMD)R_(c) ^(f), replacing the front cluttercovariance matrix R_(c) ^(f) in (2.12); and b) S_(CCPC) ^(UCMD) is theset of UCMD and CCPC parameters that define the specific CCPC case.

CCPC Case I

This first CCPC Case I has only one PCC pair and does not use any SAPsince θ¹=0° which corresponds to the physically implemented antennapattern of FIG. 5 which is directed towards boresight. The defining setS_(CCPC) ^(UCMD) is then given by the following expression:

S _(CCPC) ^(UCMD)={_(f)σ_(c,i) ²,θ¹=0°,PCNR₁=57 dBs,PCNR₂75 dBs}  (3.9)

CCPC Case II

This second CCPC Case II has three PCC pairs. One is associated with theantenna pattern of FIG. 5 and the other two with two different SAPs. Thedefining set S_(CCPC) ^(UCMD) is given by the following expression:

S _(CCPC) ^(UCMD)={_(f)σ_(c,i) ²θ¹=−7°,θ²=0°,θ³=7°,PCNR¹=57 dBs,PCNR₂=75dBs}  (3.10)

CCPC Case III

This third CCPC Case III has five PCC pairs. One is associated with theantenna pattern of FIG. 5 and the other four with four different SAPs.The defining set S_(CCPC) ^(UCMD) is given by the following expression:

S _(CCPC) ^(UCMP)={_(f)σ_(c,i)²,θ¹=−14°,θ²=−7°,θ³=0°,θ⁴=7°,θ⁵=14°,PCNR₁=57 dBs,PCNR₂=75 dBs}  (3.11)

In FIGS. 14 a-14 d the simulation results for range bin #1 of FIG. 8 arepresented for the above three cases, as well as the non knowledge-aidedSPT sample matrix inverse (SMI) scheme described below

$\begin{matrix}{w = {\left\lbrack {\,^{smi}R} \right\rbrack^{- 1}s}} & (3.12) \\{{\,^{smi}R} = {{\frac{1}{Lsmi}{\sum\limits_{i = 1}^{Lsmi}{X_{i}X_{i}^{H}}}} + {\sigma_{diag}^{2}I}}} & (3.13)\end{matrix}$

where X_(i) denotes radar measurements from range bins close to therange bin under investigation, Lsmi is the number of measurement samplesand σ² _(diag)I is a diagonal loading term. Xi was derived via thefollowing generating equation

X _(i) =R _(i) ^(−1/2) x _(i)  (3.14)

where: a) xi is a zero mean, unity variance, NM dimensional complexrandom draw; and b) Ri is the total disturbance covariance (2.11)-(2.12)associated with the i-th range bin. For the example presented herein,σ_(diag) ²=10. For the results shown in FIG. 14, Lsmi=512 correspondingto 8 passes of the 64 range bins SAR image of FIG. 8. In addition, theradar and environmental conditions parameters assumed for all thesimulations are given in Table 1 for ease of reference (note that nojammers are assumed in the simulations, however, it should also be keptin mind that outstanding SINR radar performance results are derived whenthere are jammers present).

FIGS. 14 a-14 d are now explained in some detail. In FIG. 14 a, theideal front clutter average power p_(c) ^(f)(θ_(c) ^(i),θ_(t)) of (2.15)is plotted versus the range bin cell position. Note from FIG. 5 thatrange bin cell position 1 corresponds to −90°, 128.5 to 0° and 256 to+90° where all the angles are measured from boresight. Furthermore, theaverage power axis has been marked with the corresponding CNR of 59 dBand the cell position axis with the corresponding CCC of 104.1 which isalso noted to reside 24.4 range bin cells away (−17.1°) from the assumedtarget location of 128.5 or 0°. The ideal clutter waveform is thencontrasted with the predicted ones derived from (3.4) and linked to theselected PCC for each CCPC scheme.

From FIG. 14 a it is first noted how the front clutter average powerp_(c) ^(f)(θ_(c) ^(i),θ_(t)) varies in dBs with respect to range bincell position (note from FIG. 1 that range bin cell position 1corresponds to −90°, 128.5 to 0° and 256 to +90°, all angles measuredfrom boresight). In FIG. 14 b the optimum and SMI SINR plots aredisplayed versus normalized Doppler. In FIG. 14 c the SMI adaptedpattern is given in dBs along the front clutter ridge which is describedas follows

AP(θ_(c) ^(i),θ_(AAM),β,θ_(t) ,f _(D) ^(t))=10 log₁₀ |w ^(H) c_(f)(θ_(c) ^(i),θ_(AAM))|²  (3.15)

where θ_(AAM)=2°, β=1, θ_(t)=0, f_(D) ^(t)=0, In FIG. 14 d, theeigenvalues in dBs of the total disturbance covariance R are presentedversus eigenvalue index for both the optimum and SMI schemes.

FIG. 22 is a plot of SMI-AASE as a function of the ratio of SMI samples,Lsmi, over the number of STAP degrees of freedom NM. From this figure itis noted that this ratio must be equal to 20 (corresponding to 5,120 SMIsamples), to achieve an AASE value of 3 dB which is, at least, a factorof 10 larger than that required if the SAR image had been of ahomogeneous terrain. From this figure it is concluded that the derivedSINR radar performance is not satisfactory for the SMI algorithm.

Referring now to FIG. 14 a, the legend “Pred Clutter I (8, 75.0 dB)”pertains to the front predicted clutter average power (3.4) for CCPCCase I. To understand the meaning of the ordered pair (8, 75.0 dB),reference is made to FIG. 15, which presents the front antenna patternof FIG. 5 plotted in more detail as a function of cell location for anyrange bin. From this figure it is noted that there are 15 lobes (sincethe assumed number of antenna elements is N=16; see Table 1) where lobe8 corresponds to the main lobe. In addition, this figure ischaracterized by the following set of zero crossings and mainlobe peakpositions across the range-bin:

$\begin{matrix}\begin{matrix}{I = \begin{bmatrix}{{ZP}^{1},{ZP}^{2},{ZP}^{3},{ZP}^{4},{ZP}^{5},{ZP}^{6},{ZP}^{7},{ZP}^{8},} \\{{ZP}^{9},{ZP}^{10},{ZP}^{11},{ZP}^{12},{ZP}^{13},{ZP}^{14},{ZP}^{15}}\end{bmatrix}} \\{= {\begin{bmatrix}{42,59,73,85,97,108,118,128.5,} \\{139,149,160,172,184,198,215}\end{bmatrix}\mspace{14mu} {cell}\mspace{14mu} {position}}} \\{= {\begin{bmatrix}{{- 60.5},{- 48.5},{- 38.7},{- 30.2},{- 21.8},{- 14},} \\{{- 7},0,7,14,21.8,30.2,38.7,48.5,60.5}\end{bmatrix}\mspace{14mu} {degrees}}}\end{matrix} & (3.16)\end{matrix}$

This set is then used to denote the possible directions to which thetrue antenna pattern of FIG. 5 can be shifted. Among these possibledirections are those given in expressions (3.9)-(3.11) where SAPs aredefined for three different CCPC cases. These directions can generallybe anywhere in the specified range of cell locations from 1 to 256. Infact, numerous simulations have revealed outstanding SINR radarperformance with directions that are anywhere in between the best twoadjacent directions selected from (3.15). In other words, thesedirections have only been selected because they scan the entire rangebin from cell 1 to cell 256 and have some connection to the lobes of thetrue antenna pattern. The ordered pairs appearing in FIG. 14 a areexplained as follows. The ordered pair (8, 75.0 dB) next to the titlePred Clutter I indicate that the SAP associated with the selected PCC ofCCPC Case I of (3.9) is the physically implemented antenna pattern ofFIG. 15 where the predicted clutter to noise ratio or PCNR is 75.0 dB.As a second example it is noted that the legend Pred Clutter II (7, 75.0dBs) indicates that the plotted predicted clutter covariance powerwaveform corresponds to that of CCPC Case II of (3.10) where the antennapattern had been shifted to −7° away from boresight and the PCNR is onceagain 75.0 dB.

In FIG. 14 b the SINR results derived with each scheme are presented.The title for each legend is self explanatory, and the ordered pairseach indicate the maximum SINR error followed by the average SINR error.It should be noted that significantly better results are derived forCCPC Cases II and III than the SMI case and the CCPC Case I.Furthermore, it should be noted that CCPC Case III outperforms CCPC CaseII by a relatively small amount. In FIG. 14 c the adapted patterncorresponding to all contrasted cases is plotted. The adapted pattern isdescribed as follows

AP(θ_(c) ^(i),θ_(AAM),β,θ_(t) ,f _(D) ^(t))=10 log₁₀ |w ^(H) c^(f)(θ_(c) ^(i),θ_(AAM))|²  (3.17)

where θ_(AAM)=2°, β=1, θ_(t)=0, f_(D) ^(t)=0

Finally, in FIG. 14 d the eigenvalues in dBs of the total disturbancecovariance is plotted versus eigenvalue index for each case.

Referring now to FIGS. 16 a and 16 b, the average and maximum SINRerrors are plotted versus the 64 range bins of FIG. 8. The resultspresented in FIGS. 16 a and 16 b correlate with those presented forrange bin #1. In other words, it is concluded that CCPC Cases II and III(with average of average SINR error (AASE) values of 1.2 and 1.16,respectively) yield a satisfactory SINR performance while the SMI andCCPC Case I do not.

Integrated Clutter Compressor and CCP Compressor

The results that are derived when the output of the RBCC of FIG. 10 isused in conjunction with CCPC Case III defined by (3.11) with_(f)σ_(c,i) ² replaced with _(f) ^(RBCC)σ_(c,i) ² are now discussed. TheRBCC is of the predictive-transform type and compresses the SAR imagefrom 4 MB to 512 bytes. In FIG. 17 the corresponding 512 byteradar-blind PT decompressed SAR image is shown.

Referring now to FIG. 17, a 512 bytes radar-blind PT decompressed SARimage is shown. It should be noted that the amount of compression isvery significant, i.e., a factor of 8,192, since the original SAR imagewas compressed from 4 MB to 512 bytes. This PT technique outperforms insignal to noise ratio (SNR) wavelets based JPEG2000 by more than 5 dBs.Referring now to FIG. 21, the corresponding average SINR error for all64 range bins is presented An inspection of FIG. 21 reveals an AASEvalue of 5.8 dB which is unsatisfactory for a KA type technique. Asmentioned earlier, this radar-blind technique becomes much more usefulwhen the covariance processor of expressions (2.14)-(2.15) is replacedwith a new type of covariance processor, a type that is derived using anovel processor coding methodology, which is the time compression dualof space compression source coding. A radar-seeing technique is nextconsidered that yields significantly better results than that derivedwith the radar-blind technique but that requires knowledge of theantenna pattern and range bin geometry or APRBG.

Referring now to FIG. 18, for range bin 1 the RBCC clutter average poweris plotted versus clutter cell number, as well as the associatedpredicted clutter average power for CCPC Case III. In FIG. 19, theaverage SINR error is presented versus all the 64 range bins where theAASE is given by 1.27 dB. It was found that when the conventional CCPwas implemented with the RBCC scheme it yielded an AASE value of 5.8dBs.

Finally, it should be noted that when the radar-seeing clutter coder orRSCC scheme with a compression ratio of 8,192 is combined with CCPC CaseIII, very close results to those obtained with the radar-blind case wereobtained. As a result, it is concluded that the radar-blind scheme ispreferred since besides being rather simple in its implementation itdoes not require any knowledge of the radar system where it will beembedded.

The examples presented in accordance with the present inventiondemonstrate that a SAR imagery clutter covariance processor appearing inKA-AMTI radar can be replaced with a fast clutter covariance processorresulting in outstanding SINR radar performance while processing clutterthat had been highly compressed using a predictive-transform radar-blindscheme. The advanced fast covariance processor is a lossy processorcoder that inherently arises as the time compression processor codingdual of space compression source coding. Since a more complexradar-seeing scheme generally did not significantly improve the resultsobtained with the radar-blind case, the radar-blind clutter compressionmethod is preferred due to its simplicity and universal use with anytype of radar system. In addition, since the fast clutter covarianceprocessor output departed sharply from that of the significantly sloweroriginal clutter covariance processor, it is established that whendesigning a fast clutter covariance processor for a radar application itis unnecessary to be concerned about how well the output of the fastprocessor matches that of the slower original clutter covarianceprocessor.

The emphasis before was in how well the fast signal processor outputmatches that of the slow original signal processor; however, now theemphasis is on how well the fast signal processor impacts theperformance of the overall system. The approach of the present inventionmay also be utilized in more advanced 3-D scenarios.

A fundamental problem in source coding is to provide a replacement forthe signal source, called a source coder, characterized by a rate thatemulates the signal source entropy. This type of source coder islossless since its output is the same as that of the signal source suchas is the case with Huffman, Entropy, and Arithmetic coders. Anotherfundamental problem in source coding pertains to the design of lossysource coders that achieve rates that are significantly smaller than thesignal source entropy. These solutions are linked to applications wherethe local signal to noise ratio (SNR) does not have to be infinite, oralternately, the global performance criterion of the application at handis not the local SNR. An example of the latter is when syntheticaperture radar (SAR) imagery is compressed for use in knowledge-aided(KA) airborne moving target indicator (AMTI) radar. To address the lossysource coding problem, many techniques have been developed including thestandards of JPEG, MPEG, wavelets based JPEG2000, andpredictive-transform (PT) source coding.

Lossy PT source coding, in particular, is a source coding technique thatis derived by combining predictive source coding with transform sourcecoding using a minimum mean squared error (MMSE) criterion subjected toappropriate implementation constraints. A byproduct of this unifyingsource coding formulation is coupled Wiener-Hopf and eigensystemequations that yield the prerequisite prediction and transformationmatrices for the PT source coder. The basic idea behind the PT sourcecoder architecture is to trade off the implementation simplicity of asequential predictive coder with the high speed of a non-sequentialtransform coder. Simplified decomposed PT structures are noted to arisewhen signals are symmetrically processed. A strip processor is anexample of such processing. Furthermore, cascaded Hadamard structuresare integrated with PT structures to accelerate the on-line evaluationof the necessary products between a transform or predictor matrix and asignal vector.

As shown and described herein, the excellent space compression achievedwith lossy PT source coding is not affected by its integration with avery fast and simple bit planes methodology that operates on thequantized coefficient errors emanating from the PT encoder section. Theefficacy of the methodology will be illustrated by compressing SARimagery of KA-AMTI radar that is subjected to severely taxingenvironmental disturbances. In particular, it is found that PT sourcecoding with bit planes significantly outperforms wavelets based JPEG2000in terms of local SNR as well as global SINR radar performance.

Referring now to FIG. 23, the overall PT source coder architecture isshown. It has as its input the output of a signal source y. As anillustration, this output will be assumed to be a real matrixrepresenting 2-D images. The structure includes two distinct sections.In the upper section, the lossy encoder and associated lossy decoder aredepicted while in the lower section the lossless encoder and decoder areshown. Before the lossless section of the coder is explained, whichcontains the offered bit planes, the lossy section will be reviewed. InFIG. 24, the lossy PT encoder structure is shown. It includes atransform pre-processor f_(T)(y) whose output x_(k) is a real ndimensional column vector. In FIG. 25 an image coding illustration isgiven where y is a matrix consisting of 64 real valued picture elementsor pixels and the transform pre-processor produces sixteen n=4dimensional pixel vectors {x_(k):k=1, . . . , 16}. The pixel vectorx_(k) then becomes the input of an n×n dimensional unitary transformmatrix T. The multiplication of the transform matrix T by the pixelvector x_(k) produces an n dimensional real valued coefficient columnvector c_(k). This coefficient, in turn, is predicted by a real ndimensional vector ĉ_(k/k−1). The prediction vector ĉ_(k/k−1) is derivedby multiplying the real m dimensional output z_(k−1) of a predictorpre-processor (constructed using previously encoded pixel vectors, asdiscussed below), by a m×n dimensional real prediction matrix P. A realn dimensional coefficient error δc_(k) is then formed and subsequentlyquantized yielding δĉ_(k). The quantizer has two assumed structures. Oneis an “analog” structure that is used to derive analytical designexpressions for the P and T matrices and another is a “digital”structure used in actual compression applications. The analog structureallows the most energetic elements of δc_(k) to pass to the quantizeroutput unaffected and the remaining elements to appear at the quantizeroutput as zero values, i.e.,

$\begin{matrix}{{\delta {{\hat{c}}_{k}(i)}} = \left\{ \begin{matrix}{{\delta c}_{k}(i)} & {{i = 1},\ldots \mspace{14mu},d} \\0 & {{i = {d + 1}},{\ldots \mspace{14mu} n}}\end{matrix} \right.} & (4.1)\end{matrix}$

The digital structure multiplies δc_(k) by a real and scalar compressionfactor ‘g’ and then finds the closest integer representation for thisreal valued product, i.e.,

δĉ_(k) =└gδc _(k)+1/2┘  (4.2)

The quantizer output δĉ_(k) is then added to the prediction coefficientĉ_(k/k−1) to yield a coefficient estimate ĉ_(k/k). Although other typesof digital quantizers exist, the quantizer used here (4.2) is simple toimplement and yields outstanding results. The coefficient estimateĉ_(k/k) is then multiplied by the transformation matrix T to yield thepixel vector estimate {circumflex over (x)}_(k/k). This estimate is thenstored in a memory which contains the last available estimate ŷ_(k−1) ofthe pixel matrix y. It should be noted that the initial value forŷ_(k−1), i.e., ŷ₀, can be any reasonable estimate for each pixel. Forinstance, since the processing of the image is done in a sequentialmanner using prediction from pixel block to pixel block, the initial ŷ₀can be constructed by assuming for each of its pixel estimates theaverage value of the pixel block x₁. FIG. 26 shows for the illustrativeexample how the image estimate at processing stage k=16, i.e.,ŷ_(k−1)ŷ₁₅, is used by the predictor pre-processor to generate the pixelestimate predictor pre-processor vector z₁₅. Also note from the samefigure how at stage k=16 the 4 scalar elements (ŷ₅₇, ŷ₆₇, ŷ₇₇, ŷ₈₇) ofthe 8×8 pixel matrix ŷ₁₅ are updated making use of the most recentlyderived pixel vector estimate {circumflex over (x)}_(15/15).

The design equations for the T and P matrices are derived by minimizingthe mean squared error expression

E[(x_(k)−{circumflex over (x)}_(k/k))^(t)(x_(k)−{circumflex over(x)}_(k/k))]  (4.3)

with respect to T and P and subject to three constraints. They are:

1) The elements of δc_(k) are uncorrelated from each other.

2) The elements of δc_(k) are zero mean.

3) The analog quantizer of (4.1) is assumed.

After this minimization is performed, the following coupled Wiener-Hopfand Eigensystem design equations are derived:

$\begin{matrix}{{P = {\left\lbrack {I_{m}\mspace{14mu} 0_{{mx}\; 1}} \right\rbrack J\; T}},} & (4.4) \\{{\left\{ {{E\left\lbrack {x_{k}x_{k}^{t}} \right\rbrack} - {\left\lbrack {{E\left\lbrack {x_{k}z_{k - 1}^{t}} \right\rbrack}{E\left\lbrack x_{k} \right\rbrack}} \right\rbrack J}} \right\} T} = {T\; \Lambda}} & (4.5) \\{J = {\begin{bmatrix}{E\left\lbrack {z_{k - 1}z_{k - 1}^{t}} \right\rbrack} & {E\left\lbrack z_{k - 1} \right\rbrack} \\{E\left\lbrack z_{k - 1}^{t} \right\rbrack} & 0\end{bmatrix}^{- 1}\begin{bmatrix}{E\left\lbrack {z_{k - 1}x_{k}^{t}} \right\rbrack} \\{E\left\lbrack x_{k}^{t} \right\rbrack}\end{bmatrix}}} & (4.6)\end{matrix}$

where these expressions are a function of the first and second orderstatistics of x_(k) and z_(k−1) including their cross correlation. Tofind these statistics the following isotropic model for the pixels of ycan be used:

E[y_(ij)]=K,  (4.7)

E[(y _(ij) −K)(y _(i+v,j+h) −K)=(P _(avg) −K ²)ρ^(D)  (4.8)

ρ=E[(y _(ij) −K)(y _(i,j+1) −K)]/(P _(avg) −K ²)

D=√{square root over ((rv)² +h ²)}  (4.9)

where v and h are integers, K is the average value of any pixel, P_(avg)is the average power associated with each pixel, and r is a constantthat reflects the relative distance between two adjacent vertical andtwo adjacent horizontal pixels (r=1 when the vertical and horizontaldistances are the same).

In FIG. 27 the lossy PT decoder is shown.

Bit Planes

The general architecture of the lossless PT encoder is shown in FIG. 28,which has as input the digitally quantized coefficient error sequence{δĉ_(k): k=1, . . . , N_(B)} where N_(B) is the total number ofcoefficient error vectors needed to encode the 2-D image y. The outputof the lossless PT coder is the desired bit stream {b_(j)ε(0,1): j=1, 2,. . . , N_(b)} where N_(b) is the number of bits generated by thelossless PT encoder prior to its further encoding using a losslesssource coding scheme such as an Arithmetic coder. The coefficient errorsequence forms what is called in the figure PT Blocks which is a matrixof dimension n×N_(B). In FIG. 29, an illustrative example is presentedwhere n=6 and N_(B)=6. The most energetic element of each quantizedcoefficient error is found in the first row of PT Blocks, i.e., in therow {−3 0 0 −1 1 2}, and the least energetic one is found in the lastrow, i.e., the row {0 0 0 −1 0 0}.

The PT Blocks are then decomposed into NZ_Amplitude_Locations andNZ_Amplitude_Values. NZ_Amplitude_Locations is an n×N_(B) dimensionalmatrix that conveys information about the location of the nonzero (NZ)amplitudes found in PT Blocks. From the simple example of FIG. 29, it isnoted that all nonzero elements of PT Blocks are replaced with a 1.NZ_Amplitude_Values, on the other hand, retain the actual values of thenonzero amplitudes. In FIG. 29, these amplitudes are shown for theexample where it is noted that the number of elements in each row is notconstant and also that no elements are displayed corresponding to thefourth row of PT Blocks since this row is made of zero values only.

Referring now to FIG. 28, it is noted that the NZ_Amplitude_Locationsmatrix is now split up into a Boundary matrix and a LocBitPlane block.The Boundary matrix is associated with the location where the zero runsbegin in the direction from top to bottom of each column of theNZ_Amplitude_Locations matrix. LocBitPlane, on the other hand, are thebits that remain after the 1's followed by zero runs of the Boundarymatrix are eliminated from the NZ_Amplitude_Locations matrix.

In FIG. 30, this decomposition is illustrated for the running example.It should be noted that the Boundary matrix has three symbols. They are0, 1 and X. The symbol X is used for the elements of a row whose valuesare all zero, and thus it informs about a zero row. The symbol 1 doesnot appear more than once for each column and specifies a boundarylocation where the zero run begins for that particular column. Forexample, since the zero run starts at row 4 for the first column, the 1is placed on the third row just prior to the beginning of the zero run.The aforementioned LocBitPlane is also illustrated in FIG. 30. It shouldbe noted how for the third column only the bits {0 1 1} are listed andthe zero for the fourth row is ignored since this information isavailable from the encoding of the Boundary matrix.

Referring now to FIG. 28, it is noted that the Boundary matrix isdecomposed into three blocks. They are the blocks ZeroRows,BndryBitPlane and RowOneOnes. This decomposition is best explained withthe illustrative example of FIG. 31. From this figure it is noted thatZeroRows assigns a 0 to a row of the Boundary matrix if it is composedof the special symbol X, otherwise it assigns a 1 to the row.BndryBitPlane is the same as Boundary matrix except that all rows madeup of the special symbol X are removed. In addition, BndryBitPlanereplaces a 0 with a 1 in the first row of a column with a full zero run.See for example the second column of the Boundary matrix which has afull zero run and for which a 1 has been placed on the first row of thecolumn. Finally, RowOneOnes keeps track of the ones in the first row ofBndryBitPlane that arose from replacing a 0 with a 1 as mentionedearlier. This completes the encoding of the NZ_Amplitude_Locationsmatrix of FIG. 28 into bit planes. Next the same is accomplished withthe NZ_Amplitude_Values block of FIG. 28 which was illustrated in FIG.29.

From FIG. 28 it is noted that NZ_Amplitude_Values is decomposed into twoblocks. One is a Magnitude block and the other is a SignsBitPlane block.The nature of these two blocks is illustrated in FIG. 32, where theSignsBitPlane block assigns a zero to a negative integer value and a oneto a positive integer value. The Magnitude block is self explanatory.Returning to FIG. 28 it is noted that the Magnitude block is decomposedinto X MagBitPlane blocks. Each of these component blocks are readilyexplained via the illustrative example of FIG. 33. It is first notedthat since the maximum integer value for the Magnitude block is 3 therewill be 3−1=2 MagBitPlane blocks (it should be noted, however, that ifthe integer value 2 did not appear in the Magnitude block only oneMagBitPlane block is needed with this information sent to the decoder asoverhead). MagBitPlane-1 is noted from FIG. 33 to assign a 1 to theinteger of magnitude 1 and a 0 to the other cases. On the other hand,MagBitPlane-2 ignores all integers with a magnitude of one, and assignsa 1 to the integers with a magnitude of 2 and a 0 to the remainingintegers. At this point, there are the necessary stream of ones andzeros that can then be appropriately encoded using a lossless encodersuch as an Arithmetic encoder whose output is then sent to the losslessPT decoder.

Referring now to FIG. 34, the lossless PT decoder is shown whichreceives as input the output of the lossless PT encoder (note it isassumed here that a lossless decoder such as an Arithmetic decoder wasappropriately used to derive this input). The front part of the decoderconstructs an n×N_(B) matrix, ZeroRows_M, made up of either unity rowsor zero rows depending on the nature of the ZeroRows bits. In FIG. 35this construction is illustrated with the running illustrative example.Note that the ZeroRows bits that were derived in FIG. 31 are now used toconstruct a 6×6 matrix consisting of either unity or zero rows. Next theZeroRows_M matrix is used in conjunction with the BndryBitPlane bits togenerate the n×N_(B) matrix BndryBitPlane_M. This process is illustratedin FIG. 36. The next step is to use the derived BndryBitPlane_M matrixtogether with the RowOneOnes bits to derive a RowOneOnes_M matrix thatis also of dimension n×N_(B). This process is illustrated in FIG. 37.Next the RowOneOnes_M matrix is combined with the LocBitPlane bits toderived a LocBitPlane_M matrix of dimension n×N_(B). In FIG. 38 thiscombination is shown for the illustrative example where it is noted thatthe Loc_Bit_Plane_M matrix is identical to the NZ_Amplitude_Locationsmatrix shown in FIG. 29. This rather straightforward reconstructionprocedure is appropriately continued until the desired error sequence{δĉ_(k): k=1, . . . , N_(B)} is fully derived. In the next section theproposed algorithm is applied to SAR imagery.

A Real-World Application

The efficacy of the previously advanced bit planes PT method is nowdemonstrated by comparing it with wavelets based JPEG2000 in areal-world application. The application consists of compressing 4 MB SARimagery by a factor of 8,192 and then using the decompressed imagery asthe input to the covariance processor coder of a KA-AMTI radar systemsubjected to severely taxing environmental disturbances. This SARimagery is prior knowledge used in KA-AMTI radar to achieve outstandingSINR radar performance.

The 4 MB SAR image that will be tested is given in FIG. 7. The magnitudeof the image is in dBs and consists of 1024 rows and 256 columns, andrepresents an image of the Mojave Airport in California. This image wascompressed using a 16×1 strip processor that moves on the image fromleft to right and top to bottom. In FIG. 17 the decompressed SAR imageis shown that was derived when the image was compressed by a factor of8,192 using the PT source coder of this paper. The SNR performancedescribed by

$\begin{matrix}{{SNR} = {10{\log_{10}\left\lbrack {\sum\limits_{i}{\sum\limits_{j}{y_{ij}^{2}/{\sum\limits_{i}{\sum\limits_{j}\left( {y_{ij} - {\hat{y}}_{ij}} \right)^{2}}}}}} \right\rbrack}}} & (4.10)\end{matrix}$

derived with this approach is equal to 12.5 dBs. In FIG. 39 thecorresponding decompressed image for JPEG2000 is shown. The SNRperformance for this case yields a value of 7.0 dB, which is more than 5dB away from the PT approach. In addition, the SINR radar performancederived with JPEG2000 is worse by 2 dB than that for the same PT sourcecoding technique.

In another aspect, the present invention relates to a simplifiedapproach for determining the output of a total covariance signalprocessor. Such an approach may be used, for example, in connection withan antenna-based radar system to make a decision as to whether or not atarget may be present at a particular location. Instead of estimatingthe output of a clutter covariance processor by performing certaincalculations offline, characterizing the input signal using onlinecalculations, and then using the online calculations to select one ofthe offline calculations, as discussed in the embodiments above dealingwith clutter covariance processors, in this embodiment, a single offlineset of calculations is performed and then used to estimate of the outputof the total covariance processor in conjunction with the antenna signalobtained at the time of viewing a target.

In the case of an antenna-based radar application, the antenna patternis shifted to a Shifted Antenna Pattern (SAP) and the single set ofoffline calculations are performed with this SAP in mind. The ShiftedAntenna Pattern is determined based on some determination of the cluttercentroid signal for the various range bins of the antenna image. Forexample, the Shifted Antenna Pattern may be determined based on thestandard deviation of the clutter centroid signal, or an RMS estimationof the clutter centroid signal.

In the present embodiment of the invention, it is generally unnecessaryto evaluate online the clutter centroid for each image range bin, sincethe offline calculations act to select the best global and symmetricallyplaced Shifted Antenna Pattern to use with all range bins to estimatethe total covariance signal. Although a pair of Shifted Antenna Patternsmay be used, with each pattern of the pair being symmetrically offsetwith respect to the initial antenna pattern, a single pattern of theShifted Antenna Pattern pairs may be used, with generally very similarresults. The reason as to why performance using a single pattern of theShifted Antenna Pattern pair is generally close to that obtained using apair of patterns, is generally due to the even/odd processing symmetriesof the clutter steering vectors, as well as the symmetrical structure ofthe antenna pattern.

In this embodiment of the invention, the SAP main lobe peak isrestricted to reside at only one of the five positions specified inEquation 3.11 above.

The output of the total covariance processor, or space time processor,is generally of the form:

y=x·w  (5.1)

where x is based on the antenna signal, and w is a weighting vectordetermined online. In turn, w may be determined based on the followingequations:

$\begin{matrix}{w = {\left\lbrack {\,_{CCPC}^{RBCC}R} \right\rbrack^{- 1}s}} & (5.2) \\{{\,_{CCPC}^{RBCC}R} = \left. R \right|_{R_{c}^{f} = {{}_{}^{}{}_{}^{}}}} & (5.3) \\\left. {{{}_{}^{}{}_{}^{}} \in \left\{ {{PCC}\left( {k,j} \right)} \right\}} \right|_{S_{CCPC}^{RBCC}} & (5.4) \\{S_{CCPC}^{RBCC} = \left\{ {{{}_{}^{}{}_{c,i}^{}},\theta_{shift},{{PCP}_{1} = {57\mspace{14mu} {dBs}}},{{PCP}_{2} = {75\mspace{14mu} {dBs}}}} \right\}} & (5.5) \\{\theta_{shift} \in \left\{ {{- 14^{{^\circ}}},{- 7^{{^\circ}}},0^{{^\circ}},7^{{^\circ}},14^{{^\circ}}} \right\}} & (5.6)\end{matrix}$

The SAP mainlobe peak position θ_(shift) was tested for the five casesin Equation 5.6 and the compressed/decompressed clutter source power_(f) ^(RBCC)σ_(c,i) ² was derived using a PT radar blind scheme. Whenthese five processor coders were simulated with the radar blind cluttercompressor or RBCC, the best AASE was produced when the SAP was shiftedto either 14° or −14°. More specifically, the AASE values for these twocases emulated the value of 1.27 dBs for CCPC Case III discussed above.These results suggest that the simulated test SAR image is characterizedby a pair of SAPs symmetrically placed with respect to the moving targetor initial antenna position, and where only one of the Shifted AntennaPatterns is needed to yield satisfactory radar performance. Aninvestigation of the clutter centroid information indicates that thedirection to shift the antenna pattern to over all 64 range bins may begoverned by some power of the standard deviation of the clutter centroidfrom the boresight position (CC=128.5). It is of interest to note thatwhen the clutter is homogeneous, the clutter centroid will be equal to128.5 for all 64 range bins and thus the selected SAP will point in thesame direction as the actual unshifted antenna pattern as expected.

The offline determined best angle to shift the SAP to could be used tomake the actual antenna pattern reflect this shifted position. A samplematrix inverse (SMI) technique can then be used with this SAP to yield aknowledge aided SMI scheme that can be viewed as an extreme case ofmemory space and computational time compressed KA-AMTI radar. Resultsfor an application of such an approach are shown in FIG. 40.

The SINR expressions that were used to derive the SMI-AASE results areas follows:

$\begin{matrix}{{{SINR}\left( \theta_{shift} \right)} = \frac{{w\left( \theta_{shift} \right)}^{t}{ss}^{t}{w\left( \theta_{shift} \right)}}{{w\left( \theta_{shift} \right)}^{t}{R\left( {\theta_{shift} = 0} \right)}{w\left( \theta_{shift} \right)}}} & (5.7) \\{{w\left( \theta_{shift} \right)} = {\left\lbrack {{\,^{smi}R}\left( \theta_{shift} \right)} \right\rbrack^{- 1}s}} & (5.8) \\{{{\,^{smi}R}\left( \theta_{shift} \right)} = {{\frac{1}{Lsmi}{\sum\limits_{i = 1}^{Lsmi}{{X_{i}\left( \theta_{shift} \right)}{X_{i}^{H}\left( \theta_{shift} \right)}}}} + {\sigma_{diag}^{2}I}}} & (5.9) \\{{X_{i}\left( \theta_{shift} \right)} = {{R_{i}^{{- 1}/2}\left( \theta_{shfit} \right)}x_{i}}} & (5.10) \\{{R_{i}\left( \theta_{shift} \right)} = \left. R_{i} \right|_{{G_{A}^{f}{({\theta_{c}^{t},{\theta_{t} = 0}})}}\Rightarrow{G_{A}^{f}{({{\theta_{c}^{t} - \theta_{shift}},{\theta_{t} = 0}})}}}} & (5.11) \\{\theta_{shift} \in \left\{ {{- 14^{{^\circ}}},0^{{^\circ}},14^{{^\circ}}} \right\}} & (5.12)\end{matrix}$

where: a) X_(i)(.) denotes a radar measurement from a range bin close tothe range bin under investigation; b) L_(smi) is the number ofmeasurement samples; c) σ² _(diag)I is a diagonal loading term where σ²_(diag)=10; d) x_(i) is a zero mean, unity variance, NM dimensionalcomplex random draw; e) R_(i) is the total disturbance covarianceassociated with the i^(th) range bin; and f) θ_(shift) is the angle fromboresight to where the peak of the antenna pattern has been shifted.

In the case of the example of FIG. 40, it is noted that when the SAP isshifted to either +14 or −14 degrees from boresight, a significantlybetter average of average SINR error (AASE) is derived confirming thepredicted result that was inferred from the robustness study of the CCPCCase III. Of the three SAPs simulated, the best result is derived whenthe SAP is shifted to −14° from boresight. For instance, note how forLsmi=512 the derived AASE value of 2.6 dBs is at least 3 dBs better thanthe 5.7 dBs that is derived with the non knowledge aided SMI case whenθ_(shift)=0 degrees.

Thus, the offline calculations may be performed for a single shiftedantenna pattern, and this single set of offline calculations used inconjunction with the antenna signal obtained online to determine theestimated output of the clutter covariance processor.

In yet another embodiment of the present invention, a simplifiedalgorithm for performing matrix inversion is used, for example, inconjunction with the previously described embodiment where the output ofthe total covariance processor is estimated using an inverse matrix,such as the inverse matrix R⁻¹ discussed above.

The simplified matrix inversion algorithm utilizes a sidelobe cancellerapproach for matrix inversion, in conjunction with the predictivetransform estimation approaches discussed herein. The sidelobe cancelleressentially removes and/or minimizes the effect of the antenna sidelobesignals on the antenna main beam return signal, x. Backgroundinformation relating to sidelobe cancellers may be found in J. R.Guerci, Space-Time Adaptive Processing for Radar (Artech House, 2003),the contents of which are incorporated herein by reference. The generalapproach for the simplified matrix inversion algorithm is in accordancewith Equation 5.2, which may be rewritten as:

w=(I−I+R ⁻¹)s  (6.1)

wx=Isx−(I−R ⁻¹)sx  (6.2)

where the term Isx corresponds to the main beam signal, and the term(I−R⁻¹)sx corresponds to the sidelobe signal.

Referring now to FIG. 41, therein is illustrated a block diagram of anembodiment of the space-time sidelobe canceller structure for anantenna-based radar application according to the present invention. Theinput to the sidelobe canceller is the output of the antenna, i.e.,complex vector x of dimension NM. This antenna output is multiplied bythe NM dimensional complex steering vector s of the assumed movingtarget to yield the “mainbeam” scalar complex response d₀=s^(H)x. Theantenna output x is also multiplied by a blocking (or null) matrix B(s)of dimension K×NM. The output of the blocking matrix is in turnmultiplied by the K dimensional weighting column vector w₀ given by thefollowing expression

w ₀ =[B(s)R B ^(H)(s)]⁻¹ B(s)R s  (6.3)

where R is the total covariance matrix. In turn, this multiplicationyields the scalar complex sidelobe response

{circumflex over (d)} ₀ =s ^(H) RB ^(H)(s)[B(s)RB ^(H)(s)]⁻¹B(s)x  (6.4)

which is then subtracted from d₀ to yield a scalar and complexbeamformer residue z=d₀−{circumflex over (d)}₀ whose value is then usedto determine if a target is present. The blocking matrix B(s) has K rowsthat are “approximately” orthogonal to s and given by the followingexpression

B(s)=[0_(K×1) I _(K) 0_(K×(NM−K−1))][Diag(s)T _(PT)]^(H)  (6.5)

where 0_(K×1) is a K dimensional column vector, I_(K) is a K dimensionalidentity matrix, and 0_(K×1) is a K×(NM−K−1) dimensional zero matrix.Diag(s) is an NM dimensional diagonal matrix whose diagonal elements arethe NM elements of s and T_(PT) is a NM×NM predictive-transform (PT)matrix appropriately designed making use of the PT design equation andisotropic statistics set forth below.

PT Design Equation

$\begin{matrix}{{\begin{Bmatrix}{{E\left\lbrack {x_{k}x_{k}^{t}} \right\rbrack} - \left\lbrack {{E\left\lbrack {x_{k}x_{k - 1}^{t}} \right\rbrack}{E\left\lbrack x_{k} \right\rbrack}} \right\rbrack} \\{\begin{bmatrix}{E\left\lbrack {x_{k}x_{k - 1}^{t}} \right\rbrack} & {E\left\lbrack x_{k - 1} \right\rbrack} \\{E\left\lbrack x_{k - 1}^{t} \right\rbrack} & 0\end{bmatrix}^{- 1}\begin{bmatrix}{{Ex}_{k - 1}x_{k}^{t}} \\{E\left\lbrack x_{k}^{t} \right\rbrack}\end{bmatrix}}\end{Bmatrix}T_{PT}} = {T_{PT}\Lambda}} & (6.6)\end{matrix}$

Isotropic Statistics

The first and the joint second order statistics of x_(k) and x_(k−1),i.e., E[x_(k)], E[x_(k−1)], E[x_(k)x_(k) ^(t)], E[x_(k−1)x_(k−1) ^(t)],E[x_(k)x_(k−1) ^(t)] and E[x_(k−1)x_(k) ^(t)] used in the above PTdesign equation are assumed to be stationary where x_(k) and x_(k−1),are real NM dimensional vector versions of the antenna NM dimensionalinput x at two different times. To find these stationary statistics thefollowing isotropic model is used for the samples of x_(k)=[x₁₁, . . . ,x_(1N), x₁₂, . . . , x_(N2), . . . , x_(1M), . . . , x_(NM)]:

E[x_(ij)]=K  (6.7)

E[(x _(ij) −K)(x _(i+v,j+h) −K)=(P _(avg) −K ²)ρ^(D)  (6.8)

E[(x _(ij) −K)(x _(i,j+1) −K)]/(P _(avg) −K ²)  (6.9)

D=√{square root over (v² +h ²)}  (6.10)

where v and h are integers,

is the average correlation coefficient between any two adjacent samples,K is the average value of any sample, and P_(avg) is the average powerassociated with each sample.

The first bracketed term in Equation 6.5 essentially functions to selectthe K-most energetic columns in the NM×NM covariance matrix, therebyresulting in a K×K matrix, which is then subject to the matrix inversionprocess. Thus, instead of performing a matrix inversion on a relativelylarge size NM×NM matrix, the matrix inversion is carried out on arelatively smaller K×K size matrix. As noted from Equation 6.5, the PTsidelobe canceller is signal dependent. However, its evaluation can bereadily accelerated using parallelism. When simulated with an exemplarytest SAR image, it was found that K=31 yields outstanding SINR radarperformance.

Thus, while there have been shown, described, and pointed outfundamental novel features of the invention as applied to severalembodiments, it will be understood that various omissions,substitutions, and changes in the form and details of the illustratedembodiments, and in their operation, may be made by those skilled in theart without departing from the spirit and scope of the invention.Substitutions of elements from one embodiment to another are also fullyintended and contemplated.

1. A method for determining the output of a total covariance signalprocessor based on a received input signal, comprising the followingsteps: determining a shifted pattern based on statistical informationrelating to the input signal; shifting an input signal modifier based onthe shifted pattern; receiving the input signal; and combining thereceived input signal in accordance with the shifted pattern to therebyestimate the output of the total covariance signal processor.
 2. Amethod for determining the output of a total covariance signal processorbased on a received input antenna signal, comprising the followingsteps: determining a shifted antenna pattern based on statisticalinformation relating to range bins of the antenna signal; shifting amain lobe of the antenna pattern in accordance with the shifted antennapattern; receiving the input antenna signal; and combining the receivedinput antenna signal in accordance with the shifted antenna pattern tothereby estimate the output of the total covariance signal processor. 3.The method of claim 2, wherein the determining step is performed using aclutter centroid signal of the antenna signal.
 4. The method of claim 2,wherein the determining step is performed using an RMS estimation of aclutter centroid signal of the antenna signal.
 5. The method of claim 2,wherein the estimation of the total covariance signal processor outputis based on the determination of an inverse covariance matrix.
 6. Themethod of claim 5, wherein the determination of the inverse covariancematrix is performed using a sample matrix inversion process.
 7. Themethod of claim 6, wherein the sample matrix inversion process comprisesthe step of performing side lobe cancellation.
 8. The method of claim 7,wherein the sample matrix inversion process comprises the step ofperforming matrix inversion of a reduced dimension subset of the inversecovariance matrix.
 9. The method of claim 7, wherein the step ofperforming side lobe cancellation comprises the following steps:multiplying the antenna signal by a blocking matrix and a weightingvector to produce a side lobe response signal; multiplying the antennasignal by a steering vector to produce a main beam response signal;subtracting the side lobe response signal from the main beam responsesignal to produce a side lobe cancelled signal.
 10. A programmedprocessor for determining the output of a total covariance signalprocessor based on a received input antenna signal, comprising: aprogrammed microprocessor; a program memory device containinginstructions for causing the programmed microprocessor to perform thefollowing steps: determining a shifted antenna pattern based onstatistical information relating to range bins of the antenna signal;shifting a main lobe of the antenna pattern in accordance with theshifted antenna pattern; receiving the input antenna signal; combiningthe received input antenna signal in accordance with the shifted antennapattern to thereby estimate the output of the total covariance signalprocessor based on the determination of an inverse covariance matrix.11. The programmed processor of claim 10, wherein the program memorydevice contains additional instructions for causing the programmedmicroprocessor to perform the additional following steps: determiningthe inverse covariance matrix by performing a sample matrix inversionprocess using side lobe cancellation; wherein the step of performingside lobe cancellation comprises the following steps: multiplying theantenna signal by a blocking matrix and a weighting vector to produce aside lobe response signal; multiplying the antenna signal by a steeringvector to produce a main beam response signal; subtracting the side loberesponse signal from the main beam response signal to produce a sidelobe cancelled signal.